Is Howard Roark's potted history of architecture accurate?

Renaissance architects absolutely did copy paste from the classical world. As for the two specific points Roark makes about the Parthenon (the triglyphs and column fluting), the short answer is that we can't be sure. I don't know of any evidence on the origins of column fluting, but there's this on the triglyphs:

From A World History of Architecture by Fazio et al, p46:

The origins of the architectural orders remain obscure. Vitruvius maintains that the orders were derived from earlier architecture in wood, a material that we know was once used for temples. In Doric temples, for example, triglyphs have been seen as echoing the protective panels applied to the ends of wooden roof beams, and metopes the infill panels between them. In its stone incarnation, the wooden end grain of the beams was stylized into vertical grooves, and the blank metope panel became a place for sculpture. Recent scholarship questions this derivation, proposing instead that the orders developed from a monumental decorative style using molded terracotta details, with no particular reference to structural features in wood. Even after the walls and columns were built in stone, wooden beams continued to be used for framing the roof, but these have not survived.

Diagram with legend

Recent archaeological work described in Constructing the Ancient World: Architectural Techniques of the Greeks and Romans by Carmelo G Malacrino, on a temple on the island of Naxos which was rebuilt or expanded at least three times over the course of about two centuries (early 8th to mid-6th centuries BCE) speaks to this question. Marble and other stone was introduced incrementally. The 7th century temple had wooden columns sitting on marble bases and "probably" marble capitals. This might be an indication of Naxos growing/getting wealthier, but perhaps also that stone construction involves a great deal of technology in quarrying, transport, dressing, and fixing that was lost with the fall of Minoan and Mycenaean civilisation, and had to be gradually rediscovered. So perhaps timber construction techniques were translated into marble because of piecemeal technological and logistical improvement rather than by stylistic choice or due to a lack of imagination.

To speak to Roark's general point: he is expressing a modernist's disdain for historicist architecture. The idea that appeals to tradition were acceptable/laudable justifications for design until modernism rendered them bunk by insisting that form follow function, is generally accepted (at least, as a neat way to introduce modernism). The purest expression of this sentiment is probably Adolf Loos' essay, "Ornament and Crime" (1913).

Of course, architecture in some periods was more conservative and likely to contain vestigial stuff than others; some vestigial stuff doesn't preclude innovation or pragmatism in other areas; and architects described as modern made and do make appeals to tradition in principle if not in form, or will pick formal precedents from the remote past or other cultures.

For example, modernist architects were kinder to traditional architectural styles which aligned with their minimalist ethic/aesthetic and ideas about modularity. The Japanese Sukiya style is an example. Walter Gropius, the founder of the Bauhaus School, wrote this to Le Corbusier on a postcard from Kyoto (I believe in 1954):

Dear Corbu, all what we have been fighting for has its parallel in old Japanese culture. This rock garden of Zen-monks in the 13th century—stones and raked white pebbles—could be by Arp or Brancusi—an elating spot of peace. You would be as excited as I am in this 2000 year old space of cultural wisdom! The Japanese house is the best and most modern I know of and truly pre-fabricated. Hoping you are well. Greetings to you and Mme. Yours—Gropius

Even the Parthenon, which Roark uses as an example of narrow appropriation, was to Le Corbusier a "product of selection applied to an established standard" (though his reasons for believing this are barely stated). Photos of the Parthenon were scandalously juxtaposed with those of automobiles in his still influential modernist manifesto, Toward a New Architecture (1923).

Almost all architects, modernist or not, regard the Parthenon as great architecture. Though I do remember reading a pithy quote about how the Greeks, to whom we owe so much philosophy, wasted their ingenuity in the field of architecture on the articulation of column capitals. I wish I could remember who wrote/said it.