Short Answer: The modern cocktail--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 cocktail. 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 cocktail 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 cocktail.
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 cocktail 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 basement 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 cocktail 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 cocktail 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 cocktails" predate Prohibition. In fact, the birth of modern cocktail 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 "cocktail," but Wondrich speculates it has its origins in Connecticut or western Massachusetts in the late 18th/early 19th century (Wondrich, 291).
