Was "the modern cocktail" intended to mask the foul taste of Prohibition-era liquor?

Short Answer: The modern c**ktail--as defined by OP as including sweetener, ice, and decorations--predated Prohibition. While Prohibition-era bartenders did need to mask bad liquor, these techniques were not influential because they were only necessary in the presence of bad liquor. The long-term effect of Prohibition was to introduce American bartenders to European bartending techniques and liquors.

Sugar: Sugar has always been at the heart of the c**ktail. I won’t reproduce individual recipes, but "loaf sugar" or simple syrup is called for in nearly every recipe listed in David Wondrich’s Imbibe, which is a history of pre-Prohibition c**ktail culture in the United States. These recipes also often call for sweeteners such as berries.

Ice: “Iced drinks had always been available for the few, but in the 1830s, with the burgeoning trade in fresh, clean New England ice, delivered by horse-drawn carts from insulated central warehouses even in the hottest months of the year, ordinary people started getting used to the stuff, expecting it, calling for it in their drinks” (Wondrich, 40).

Showmanship: Showmanship also long predates Prohibition. Here is an excerpt from an 1853 article on the New York barkeeper George Augustus Sala:

The bar keeper is . . . an accomplished artist . . . [of] unapproachable skill in compounding and arranging these beverages, and making them not only exquisite to the taste, but delightful to the view. His drinks are pictures . . . The barkeeper and his assistants possess the agility of acrobats and the prestidigitative skills of magicians. They are all bottle conjurors.—They toss the drinks about; they throw brimful glasses over their heads; they shake the saccharine, glacial and alcoholic ingredients in their long tin tubes . . . (Wondrich, 25-26)

Note also the mention of ice and sugar (“glacial” and “saccharaine”), indicating again that sugar and ice have long been integral parts of a c**ktail.

Pyrotechnics

One notable piece of 19th century showmanship was the “Blue Blazer,” pictured here:

Garnish: Exotic garnishes were not new to the Prohibition era. Late nineteenth-century barkeepers used citrus twists, berries, pickled French hazelnuts, pickled walnuts, olives, pimolas—and of course, the maraschino cherry (Wondrich, 53).

Fancy Glassware: Definitely pre-Prohibition. There's a great picture of fancy stemware from 1905 in Wondrich's book (p. 55) but I'm unable to grab it from Google Books. The coupe, the punch glass, the sherry glass, the highball, the goblet: they're all there.

Straws: Dentists thought that ice was very bad for your teeth, so the advent of ice in 1830 is also when straws were added to the c**ktail repertoire (Wondrich, 40).

Umbrellas and Tiki Drinks: Although drinks loaded with orgeat and exotic juices are able to mask indifferent liquor, these are post-Prohibition developments as well. They are associated with the experience of American servicemen in the South Pacific and were popularized by “Don the Beachcomber” in the 1940s.

So What Effect Did Prohibition Have?

It is definitely true that Prohibition increased the prevalence of bad liquor, and that bartenders had to adjust their techniques accordingly:

The stuff that was fueling this binge wasn't very good. In the cheaper joints, it was rotgut distilled in a bas*m*nt somewhere by folks who had no care for such fine points as taste, quality or even elementary sanitation. In a few of the most expensive places, joints such as New York's 21 Club, you could, for a price, get some of the whiskey, rum and champagne that was being smuggled in from abroad in staggering quantities, and if you were very, very lucky, it would be uncut. If you were a little less lucky, it would be cut responsibly with filtered water and grain alcohol . . . If you weren't lucky, of course, at best you'd get rooked completely and at worst you'd end up in the hospital or even the morgue. Prohibition, in short, was not a good time for the discriminating drinker, or for the mixologist. The bartender's art was directed towards covering the taste of bad liquor, not enhancing the taste of good. (Wondrich on Prohibition)

But “true” bartenders wanted to innovate, not mask bad liquor, so they left the United States for Europe:

Up to this point, the European school of mixology was a mere sapling in the shadow of the mighty American oak. With the Volstead Act, Congress chopped down the oak. Fancy drinking on the American plan was no longer possible in America. Not even the best, most honest speakeasies (and there were a few) could secure all the stuff the dedicated mixologist needed to practice his craft, although some did remarkably well. As for the standard-issue speak…one shudders to think.

American emigration had a big effect on the future of American c**ktail culture, in part because it led to the publishing of the enormously influential Savoy Cocktail Book:

Others, bartenders-in-their-souls and artists like Al, chose rather to exercise their art in exile than abandon it or practice it surreptitiously and with adulterate materials. Among them was Harry Craddock . . . In 1920, he got a job in London at the Savoy Hotel's American Bar . . . Six years later, he published what he had learned, in the form of the instant classic Savoy Cocktail Book. Not only were the standard American drinks of the pre-Prohibition era there, but it was jammed with drinks that had had never slid across an American bar. Drinks based on strange, foreign aperitifs such as the French Kina Lillet and Quinquina or the South African Caperitif; exotic liquors-Calvados, vodka (practically unknown in the States) and even Canadian whisky, in place of the unavailable rye and bourbon (Canadian whiskies had certainly been marketed in the States before Prohibition, but only one known c**ktail recipe from the period actually called for it); oddly-named liqueurs and unusual syrups (it's safe to say not even the Hoffman House would have carried sirop de groseille; it's made of red currants and you still can't get it here). The formulae were simple, streamlined, without the rococo refinements of composition and technique characteristic of the vanished American school. But they were elegant, too, and often imaginative. The Savoy Cocktail Book . . . [was] an ark for what was left of the American school, written at a time when Americans were marveling at European bars like the old European travelers once had at American ones.

In short the mixing techniques practiced by bartenders trying to mask the flavor of rotgut may still inspire those serving drinks at high school and college parties, but the vast majority of the refinements we associate with "modern c**ktails" predate Prohibition. In fact, the birth of modern c**ktail culture is usually ascribed to Dale DeGroff and his rediscovery of those “baroque” bartending techniques that characterized the 19th century and were lost during Prohibition.

PS: I know OP isn't asking about the origins of the word "c**ktail," but Wondrich speculates it has its origins in Connecticut or western Massachusetts in the late 18th/early 19th century (Wondrich, 291).

The Oxford English Dictionary attests the use of c**k-tail as a mixed drink from 1809 in W. Irving's Knickerbockers:

They lay claim to claim to be the first inventors of those recondite beverages c**k-tail, stone-fence, and sherry-cobbler.

and from 1839 c**ktail as a more general mixed drink in Marryat's Diary American:

He frequents the bar, calls for gin c**ktails, chews tobacco and talks politics.

Not much need for bootlegging that early in the 19th century. Whether the popularity of c**ktails in the 1920's arose from the need or desire to hide foul liquor I cannot say; but the word was long established by that time.

The Manhattan is attested to from William Schmidt's "The Flowing Bowl", published in 1891. In it, he details a drink containing:

• 2 dashes of gum (gomme syrup),
• 2 dashes of bitters,
• 1 dash of absinthe,
• 2/3 portion of whiskey and
• 1/3 portion of vermouth.

The Martini is attested to from the 1888 Bartender's Manual, where there was a recipe for a drink that consisted of:

• half a wine glass of Old Tom Gin and
• half a wine glass of vermouth.

Going further back, we find How To Mix Drinks, or The Bon-Vivant's Companion, published 1862, complete with numerous popular c**ktails from the age of Lincoln and the Civil War:

When our country's very first bartender's guide, How to Mix Drinks, was published the year after Abraham Lincoln took office in 1861, America entered a golden age of c**ktails. As it turns out, a lot of what was available to drink back in the 16th president's day has now come back in fashion

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This wealth of evidence for the popularity of both sweetened liquor drinks, and their appellation as c**ktails, predates the the Roaring Twenties and Prohibition by several decades.