You ask, “How much grammatical leeway can advertisements enjoy?” The answer is as much grammatical leeway they can get away with in this democratic, laissez-faire society of ours, and as serious a language travesty the buying public can take before they make their collective outrage heard loud and clear by the advertisers. Sadly, there’s no law or legislation against bad grammar and usage in advertising, so advertising messages are totally at the mercy of the exigencies of strong recall, as in “tomatoest” for “the most tomato-tasting” product.
Now, you also ask, would I continue patronizing that product or believe its endorser for making that “tomatoest” claim? If the product is good for its price, I certainly would. The “tomatoest” claim isn’t enough cause for me to boycott the product. We must keep in mind that a product is distinct from its advertising. A manufacturer makes a product to satisfy a need or desire, but advertising seeks to catch attention and establish strong recall for that product. So, in a highly competitive market saturated with thousands of advertising messages, product advertisers spend fortunes and often flaunt grammar rules just to get their product messages heard above the din. There ought to be a law against that, but there isn’t.
As to that school banner’s “one-on-one” computer claim, it’s a slippery and grammatically flawed way of saying “We provide one computer for every kid who enrolls in our school,” but I think it gets its message across very concisely and rather effectively. Advertising headlines and advertising copy often tend to take liberties with language that way—sometimes deliberately mangling grammar just to get attention. But were you right for not letting your cousin study in that school for that grammar travesty? I must say I don’t think so. That decision of yours was rather rash and certainly not warranted by the gravity of the offense. It’s a non sequitur—a response that does not follow logically from its premise.
Now to your last question as to which of these statements is correct: “The winner will receive major prizes” or “The winner will receive a major prize?” I would say both are grammatically correct for their different contexts. The first statement promises several major prizes while the second promises only one major prize. However, the voice-over in that commercial, “The winner will receive a major prizes,” obviously flubbed basic English grammar. As we all know, the article “a” isn’t needed when the noun is in its plural form. It’s probably simply a misreading of the script that the voice-over producer had overlooked—forgivable in such a frenzied, breakneck medium as broadcast television.