The modern theory of primary colors hadn't developed yet for the ancient world.

Interestingly (and surprisingly to a lot of people), it also appears that most of the ancients didn't consider blue a color, and didn't typically describe things like the sky or the seas as being blue.

Blue was a latecomer among colours used in art and decoration, as well as language and literature. Reds, blacks, browns, and ochres are found in cave paintings from the Upper Paleolithic period, but not blue. Blue was also not used for dyeing fabric until long after red, ochre, pink and purple.

For example, according to Pliny the Elder^(ibid) the primary colors were red, yellow, black, and white. In modern terms, yellow and magenta are two of the three primary subtractive colors that you can use for dyes, and black and white can be used as the key ("K") ink. But of course without magenta, you conspicuously can't get any blueish hues. He was apparently unaware of (or unimpressed with) that entire part of the color spectrum.

To add to TED's answer, another reason they didn't do this is that they simply outlawed wearing purple, which was the imperial color:

The Sumptuariae Leges of ancient Rome were various laws passed to prevent inordinate expense (Latin sūmptus) in banquets and dress, such as the use of expensive Tyrian purple dye.

Short answer — frame challenge:

Of course 'someone' did that, perhaps quite early on, we can quite safely assume. Although hard evidence for that is scarce and presented here only with one concrete example at the end of this answer.

But — what we can be very sure of is, that for the underlying assumptions both prima facie as well as implicit, for most of the time there was neither a need nor a direct purpose for doing so.
This is revealed upon closer inspection of the comments below the question, the word "cheap" in the title, and even more so in the question's direct subquestion in body:

Q: Also, why did the ancients not simply mix red and blue dye together to make purple?

Together with the bit from the question's title: "… before purple dye became cheap" — seems to assume that 'naturally achieved purple dye' had been always prohibitively expensive, only to become cheap when de novo chemical dyes became available, setting an implicit time-frame for this as '~before 20th century'. But this base assumption is not true.

Thus the assumptions spelled out are: 'purple dye was rare, expensive, therefore not used often and alternatives lacking'. These prerequisites are only partially true for a subset of use cases. In general they are untrue. Only murine sourced sea-shell purple, also known as 'true purple' was that expensive, and alternatives for that were numerous.

The answer to that subquestion is therefore: they did mix red and blue dye to achieve purple! And even for the expressed purpose of imitating royal purple, only much cheaper, and still very good at that. This satisfied a high demand, from many classes, and sumptuary laws or not, it was used, widely.

Thus, neither lack of knowledge, nor weaving technique limitations, nor any laws anywhere explain the 'why not' for lacking finds and evidence for this colour mix technique employed in making cloth. It could be done by interweaving, and was done, but it was much cheaper as a direct dye with much better results, if the purpose was 'purple cloth'. Cheap adequate purple dyes were not just widely available and usable but to be indistinguishable upon closest inspection. That means that interweaving most probably would have been done for most applications for the very effect the interweaving would give.

Long form:

Prolegomena, textile technology, weaving and dyeing

The questions touches a lot of aspects. Cost, law, inherent qualities of the textile colour — like shine, fastness, alternative dyes' availability. The apparent wavelength perception by human eyes for a given piece of cloth at various viewing angles and distances are not the only problem to appreciate.

It is obviously quite easy for some weave patterns — like canvas-like balanced plain weave — of fine threads to give an impression of red weft and blue warp (or vice-versa) to achieve a purple impression with increasing viewing distance:

Figures 5 and 6 are two examples of such mixtures. They were produced using many repeats in warp and weft directions, thread count close to real cloth, and assuming there are no spaces between the yarns, which is reasonable assumption for most woven fabrics. Figure 5 is the color simulation produced from red warp yarns and blue weft yarns and plain weave of Figure 4(a). While the color simulation of Figure 5 is produced from red warp yarns and blue weft yarns woven in sateen of Figure 4(c). These two examples indicate that numerous purple colors can be produced from only two colors (red and blue).

— Kavita Mathur & Abdel-Fattah M. Seyam: "Color and Weave Relationship in Woven Fabrics", in: Savvas Vassiliadis: "Advances in Modern Woven Fabrics Technology", 2011. DOI: 10.5772/20856

As any pixel pusher artist should know this effect from dithering, this is such a basic technique that it must have been known from quite early on.

Eurocentric assumptions, Judeo-Christian and Roman-Byzantine heritage

The high-status attribution with the associated costs for 'purple' we have in mind when discussing this are a heritage of the Roman and Byzantine uses of one special colour in use then: the 'sea-shell purple' from Murex species for Tyrian Purple or Royal Purple, produced in Europe since Minoan times. Other European centres of production for sea-shell purple dye were Croatia, Spain, Greece, Italy (— Beatriz Marín-Aguilera et al.: "Colouring the Mediterranean: Production and Consumption of Purple-dyed Textiles in Pre-Roman Times", Journal of Mediterranean Archaeology 31.2, 2018, p127-154. doi).

While it is true that the communication about colour perception may have changed quite significantly over time, it is certainly true that human eye rods and cones changed evolutionary very little over the time period inquired. Certainly as well, the physical and chemical characteristics of pigments and dyes did not change as to the wave-lengths they reflect, at least when new. The shades actually used of this 'true purple' are not only seemingly range from purplish-red to violet-blue.

The problem is then of course, what we perceive commonly as and call 'purple' is now perhaps more reddish or more bluish in shade or hue than the actual colour meant in more historic, possibly ancient times. Illustrated by an alternative name for Tyrian Purple: Phoenician Red.

Archaeological finds date this advanced textile techniques of colour effects to 'probably rarely used in bronze age' settings for Europe, but certainly more so for Hallstatt period onwards. A problem here is that pattern-, weave- and colour-effects may have been quite striking, but whether mineral, animal or plant based, the colours themselves together with the fabric still posed some inherent problems which seemed to have required specific solutions:

Bronze Age finds of colour patterns remain exceptions, but patterned fabrics were extremely popular in the Hallstatt period. They are no longer simply patterns with different natural shades, but colourful striped and checked designs with dyed yarns that have been combined effectively. According to recent dye analyses on the textiles from Hochdorf366 and Hallstatt367 (see chapter B5), weld was primarily used for yellow, woad for blue and orchil for red, a dye that can be produced from lichens, amongst others. Valuable, imported dyes are also among the colours used in the Hallstatt period, for instance the red colouring of the kermes scale insects, native to the Mediterranean. Various dyes and dyeing techniques were combined to achieve certain hues. Both fleece and yarns were dyed, for instance for striped or checked pieces. To achieve a h*m*geneous uniform colour for monochrome cloths and garments it was easier to use undyed yarns and treatment in dye baths after weaving.

— Karina Grömer: "The Art of Prehistoric Textile Making. The development of craft traditions and clothing in Central Europe", Veröffentlichungen der Prähistorischen Abteilung (VPA) 5 Natural History Museum Vienna, 2016. (pdf)

This shows one early problem with the inquired approach: what is the goal of mixing two differently coloured threads? Is it 'creating something purple'? Easy. Is it 'imitating royal purple'? Much more difficult.

Royal purple is a quite fast dye that has a shine to it. One advantage is that it uniformly dyes a fabric, but also changes appearance in specific ways depending on lighting conditions and viewing angles.

But that is achievable without any knowledge of the underlying chemistry or physics, just by empirical accumulation: tinkering long enough with the raw materials, if one very specific result is required or desired, and refine that tradition.

To illustrate this globally, in a very different cultural context, as the question is that broad to include "anyone":

The Mayans heated the raw materials in the range of 70–130°C (66), allowing the indigo to penetrate the palygorskite tunnels and paving the road for a series of ultrastable synthetic organic pigments in a clay host used in a variety of applications. One of these applications is the methyl red@palygorskite that uses a similar method of production except that the indigo is substituted by the organic molecule methyl red to obtain a purple-colored host/guest supramolecular complex.

— Katherine T. Faber et al.: "Looking Back, Looking Forward: Materials Science in Art, Archaeology, and Art Conservation", Annual Review of Materials Research, 51:435–60, 2021. doi

Laws, fraud, and near perfect imitations

One argument presented for 'not intwerweaving red and blue' is presented on this page as "Sumptuariae Leges" 'prohibiting excessive consumption' and cost and thus waste. That holds of course true for true royal purple:

“[R]oyal” purple … had come to be associated exclusively with the emperor. For anyone else to wear purple was tantamount to their plotting against the state. The ownership of any purple-dyed cloak or any cloth dyed with the finest purple or even an imitation of it incurred severe penalties.

[…]

Roman citizens still used the imperial color. This prompted another edict by Theodosius II in 424 AD prohibiting the general creation, ownership, and use of high quality purple—for it was a crime akin to high treason:

Nor shall any person at his home weave or make silk cloaks or tunics which have been colored with purple dye and woven with no admixture of anything else. Men shall bring forth from their homes and deliver the tunics and cloaks that are dyed in all parts of their texture with the blood of the purple shellfish. No threads dyed with purple dye shall be interwoven, nor shall threads colored by the same dye be spun out and made strong by the shrill sounding loom. Garments of all-purple must be surrendered to the treasury and must be immediately offered. … But let no man now by such a concealment incur the peril of the toils of the new constitution; otherwise he shall sustain the danger of involvement in a crime similar to that of high treason. (Pharr 1952, 28)

— Charlene Elliott: "Purple Pasts: Color Codification in the Ancient World", Law & Social Inquiry, Volume 33, Issue 1, 173–194, Winter 2008. jstor (emphasis added, LLC)

As is evident from the quote itself, a law that has to be repeated this explicitly and often reveals primarily that it was broken quite often. And it also caveats the casual reader that this is for the expensive dye used, even if formulated more generally. Why would it be a a luxurious sin or unlawful act against excessive consumption if one used the cheap imitation version of the dye? In that case and for the above law, we only see a marker of class distinction being protected for brand value.

Such a brand protection in conditions of cheap abundance may gave rise to the use case envisioned above in the question: interweave red and blue, look like class even from quite close-up, but if challenged or charged to infringe, one could easily point to close inspections and point at entirely different threads used than those covered under that law.

A law, of course, that presupposes a certain legal context and even more so: reach. What any Roman emperor or pope might declare as 'highly illegal', not only uppity citizens would likely and provenly still desire. On the other hand, another 'wild heathen' would perhaps not even know of that laws existence or simply chose to 'violate' such sumptuary prohibition just because of that.

Penelope Walton Rogers reports that lichen purple has been identified on some textiles from sixth- to seventh-century Anglo-Saxon graves, although it was confined to embroidery, narrow woven bands and accessories such as a bag and a headdress. There is also evidence that purple obtained from orchil lichens was used in early medieval Ireland. Tests carried out at Trinity College Dublin on the Book of Kells identified orcein, a dye produced from several species of lichen, on the Book of Kells. By the tenth century lichen purple had became standard among the Norse, and tests on woollen textiles of the Hiberno-Viking period from Dublin showed that the commonest colour used there was purple obtained from lichen. These results were compared with analyses of Viking age woollen textiles from Norway and England, with very interesting, if rather puzzling, results. To quote Penelope Walton Rogers:

The most surprising discovery was the way in which the dyes divide up into the different areas… a glance at [the] evidence might lead one to imagine that the well-dressed Norwegian wore nothing but blue, Dubliners wore purple, while in England madder-red was the fashionable colour.

Penelope Walton Rogers found it hard to know if these results really do represent varying tastes in colour in different area. Whatever the truth of this, it is fascinating to learn that the commonest colour of the clothing worn in Hiberno-Viking Dublin appears to have been purple obtained from lichens, which are easily collected in Ireland.

— Niamh Whitfield: "Aristocratic Display in Early Medieval Ireland in Fiction and in Fact: The Dazzling White Tunic and Purple Cloak", Peritia 27 (2016) p159–188. doi

Evidence for good and cheap imitation dyes being available

From the same period that we have records of Tyrian purple, we also have recipes for ways to “stretch” or imitate this much-desired shade. The dyeing of wool with orchil is mentioned by both Theophrastus and Dioscorides, with the comment that “when fresh, its color is so beautiful that it even excels the ancient purple of Tyre.” (I guess it depends on your taste.) Pliny also records that orchil was used as a “bottom” or initial dye for wool which could then be colored purple with a much smaller quantity of the expensive Tyrian purple. […]

The “secrets” of Tyrian purple and orchil dye seem to have been much less widely known in Europe from the fall of the Roman Empire until about 1300. However, Taylor and Walton (1983) are cited by Grierson as reporting “recent discoveries of purple lichen dyes on ancient textiles in some specialist rural weaving centres in ‘Dark Age’ Europe.” This is entirely possible, as the dye is quite distinctive chemically and can easily be distinguished from Tyrian purple or other dyes. (She also cites Digby, 1957.)

The much-quoted, and apparently well-established, story of how orchil was re-introduced to Western Europe was written by G. Marcotti and privately printed in Florence in 1881. The story is that one Ferro, or Federico, began manufacturing orchil dye in Florence in the early 1300s, having brought the knowledge of how to make it from the Levant. He seems to have been successful at keeping the secret in Florence for some time, and his family – who took the surname “Oricellari” from the dye – monopolized its manufacture for nearly a hundred years. It was traded as far as Spain, France, Germany and England, usually in the form of processed and dried lichen. […]

Several writers note that despite its fugitive nature, orchil remained popular as a supplemental dye for a very long time. It has been used to add depth, richness, “lustre” and brilliance to the duller purple shades produced with red and blue dyes. Annette Kok in her article in the Lichenologist says,“It is not only the shade but the quality of the color which seems to be unique; these dyes actually impart a softness and lustre both to silk and wool, whereas many dyes from other sources and their associated processes of mordanting are harmful to these fibres and have to be used with the greatest care so as not to produce harsh cloth and lustreless colors.”

— Christian de Holacombe: "Orchil, the Poor Person’s Purple", Tournaments Illuminated, No 139, 2001. (pdf)

From even way earlier, we see the biblical colours Argaman and Tekhelet were based on expensive, real, royal purple. And from that time we have ancient finds, historical surces and experimental archaeology results, for mixing cheap purple dye, from plant sources like madder and woad:

In most cases, the imitation dye was of such high quality that it was difficult to distinguish between textiles dyed in molluscan purple and those dyed with a much cheaper substitute—only advanced analytical methods enable identification.

— Naama Sukenik: "Counterfeiting and Fake Dyes in the Clothing Industry from the Roman Period", excerpt from: Sukenik: "2000 Years Old Counterfeiting Industry", in: Meiri O, Bloch Y, Kaplan Y (eds): "Out of the Blue". Jerusalem: Bible Lands Museume; 2018. pp. 100–113.

Or simply:

Due to the high value of true purple, it was frequently imitated by combining red and blue plant dyes or by using certain lichens. The earliest instance of purple dye use is currently documented in some mineralised textiles found at Qatna, Syria and dated to the Bronze Age.

— Margarita Gleba & Ulla Mannering: "Introduction: Textile Preservation, Analysis and Technology", ATS 11, 2012.

Archaeological evidence of red and blue fibres interwoven to give an impression of purple

One example for when viewed from a slight distance giving a purplish impression by using really red and blue fibres is found far from any Roman emperor or Roman bishop's jurisdiction:

During the last quarter of the 7th century BC, which saw the onset of the Late Hallstatt period (HaD), certain centres of power began to arise in southern Germany and neighbouring regions (Kaenel 2005). […]

The analysis further demonstrated that the preservation state of the fibre, i.e. the degree of degradation, was not decisive for the survival of dyestuffs. For instance, in some relatively well-preserved fragments, with visible red and blue traces of colour, dyestuffs could not be detected, whereas some black fragments with highly decomposed threads had definite traces of indigotin or kermesic acid. The results of the dye analyses also confirmed that the textiles in the burial chamber were predominantly red and blue. The dyes were used for single-colour fabrics, in dogtooth patterns or in rich patterns in the tablet-woven bands.

In all likelihood, the source of indigotin was woad (Walton-Rogers 1999, 243). The source of the red dyestuffs shed new light on the close ties of the early Celts with the civilizations south of the Alps. The high percentage of kermesic acid and the complete absence of carminic acid indicate that the dye source was the insect Kermes vermilio, which is to be found predominantly in the Mediterranean littoral. Its northernmost growing area is southern France. Since the examination of the textiles indicated domestic production, we can conclude that it was the dyestuff that was imported from southern Europe. Dye experiments with Kermes vermilio have demonstrated that the palette of possible hues is extensive, varying from maroon-red to purple. As with all dyestuffs that require a mordant, the preparation and the tools used will have a decisive effect on the resulting hue. In the finely woven, dogtooth patterned fabrics, the red and blue colour areas can be seen, but they merge into a purplish colour at some distance.

— Johanna Banck-Burgess: "Case Study: the Textiles from the Princely Burial at Eberdingen-Hochdorf, Germany", Margarita Gleba & Ulla Mannering (eds): "Textiles and Textile Production in Europe from Prehistory to AD 400", Ancient Textiles Series (ATS) Vol. 11, Oxbow Books: Oxford, Oakville, 2012.

Speaking as a lazy crochet artist and textile enthusiast, most people have pointed out the dye aspects regarding purple colors very well.

Meanwhile, I'll just point out that weaving blue and red threads together does not always give you purple cloth. Your average "blue" and "red" colors wouldn't blend together to make purple, like paint; you'd get blue-and-red stripes or plaid, depending on the weave.

While certain weaves for blue-and-red would seem purple, anyone who took a closer look would know the difference. And given the sumptuary laws that punished people for wearing colors above their station, you'd almost certainly get in trouble for it.

Google search: "blue red cloth woven"

EDIT: Adding my other comment to this one for a visual explanation of why "weaving red and blue threads to make purple" wouldn't work.

This video has a pretty good visual explanation as to why weaving blue/red threads wouldn't work completely. It's basically the woven equivalent of cross-hatching. Also, you wouldn't see the warp for most weaves since it's a structural part of the cloth.

If two very similar colors of thread (looks like red/orange and yellow) are still clearly distinguishable when right up against each other, you're not going to have much success with contrasting colors like blue and red.

Maybe in a plain weave (with the most basic over/under pattern), you'd use a blue warp and a red weft or the reverse, but weaving is notoriously labor-intensive in preindustrial times. That's why people mechanized it as soon as they could. If you have enough smarts to try out "mock-purple cloth," you're not going to waste a couple weeks on just doing a plain-weave of blue/red, and expecting people not to notice that you, someone who can't afford real purple cloth, are suddenly wearing a "purple" dress/tunic out of nowhere.

EDIT #2:

The question as I read it was pretty vague, and that's basically why my answer focused more on the practical side of weaving/textiles.

The "Wikipedia article on purple" mentioned by OP is most likely this one, which has an easy-to-spot reference to the "Tyrian purple" color, as this quote says: "Purple has long been associated with royalty, originally because Tyrian purple dye, made from the mucus secretion of a species of snail, was extremely expensive in antiquity."

So since everyone handled the "expensive dyes are for expensive people" aspects, I'll handle the first part of the OP's question of "if purple dye was so expensive [in Ancient Rome,] why didn't people weave blue and red threads together to fake a purple color?"

Short answer: They did, as mentioned in @LangLangC's answer. They did it so much that officials had to make laws specifically saying, "Stop tricking people, peasants! Bring your stuff in, so we can be sure you're not trying to dress above your station."

And on a practical basis, here is WHY faking people out with blue-and-red threads would be impractical. Holy Weaves' website has many lovely bolts of shot-silk fabric, where silk of two different colors is woven together. In this photo, blue and pink threads are woven together--and patches of it do seem to be a LOVELY purple.

The problems with Fake-Purple Option #1: To my knowledge, only silk has that iridescent effect, and silk is NOT affordable by your average weaver-girl or enterprising tailor. Moreover, it doesn't look purple EVERYWHERE, just where the light hits wrinkles differently. A person wearing shot-silk therefore a) is rich enough to afford Tyrian purple dye, instead of faking people out with cheaper options, and 2) is not wearing "purple shot-silk." They would be wearing "blue-and-pink shot-silk that switches colors with every movement." People have noted that shot-silk is almost holographic with how shimmery and translucent it can get, and that is... ALSO not cheap, nor easy to imitate.

Let's see what happens when you downgrade "weaving blue and pink silk" into something that's not shimmery or expensive. Warp And Weave helpfully has a color-mixer that tests out color combinations and basic weaves, so here's "blue and red" in a plain-weave simulation!

Does that look "purple?" Nope, it looks like red-and-blue checks. On my screen, it's actually looking like a magenta/pink color, so let's mess with the shades a bit.

Much better! But it's still clearly "maroon red and gray-toned blue, woven together." You could test out the "blue and red make fake-purple" cloth with EXTREMELY fine thread, and the best color-matching you can manage--but at some point, that's way more trouble than it's worth, because preindustrial weaving takes forever.

My sister and I have attempted to weave before (a backstrap loom for her, and tablet weaving for me) and it took us both two hours to set up our test-looms. The finicky nature of setting up a loom does not get much faster with experienced weavers than it does with beginners, because experienced weavers use thinner thread that needs very delicate handling, while beginners tangle up thicker yarn and have to redo a lot of things.

At a distance (arm's length? across the street?), you may certainly think the second option is "purple fabric," but canny officials would NOT stay at arm's length. They'd inspect the shit out of the "purple" fabric, as noted in LangLangC's answer. How far is your enterprising Roman commoner willing to take the "cheap purple" ruse, especially when a lot of Romans have tried it already, and got caught enough for officials to tell everyone "QUIT THAT, YOU UPPITY PEASANTS?"

And this is just focusing on the weaving. Dyeing thread/yarn however many shades would take at least a day per batch, for any properly sized garment; the SPINNERS who you keep buying from for your experiments would also wonder why you haven’t made anything practical yet, unless you’re paying them extra.

One or two women needed a huge portion of the year just to spin and weave the cloth for their own family’s needs, so one more yard of your experiments means one LESS yard of actual clothing, or bed linens, or sackcloth. Most preindustrial women would not have the time or energy to spend on experiments like this, and the ones that DID were obviously caught.