Advertising agencies often take liberties with language, of course, so the grammatically flawed construction "It's time everyone flies" probably could be excused for its violation, like the classic cigarette slogan "Tastes good like a cigarette should" (which, formally, should be "Tastes good as a cigarette should). You see, in business, you couldn't argue with success. But you're right; the formally correct construction for that airline slogan is "It's time everyone flew," as in "It's time everyone kissed" instead of "It's time everyone kisses." The excruciatingly correct form for that sentence is, in fact, the complex sentence "It's time that everyone kissed." In this case, however, the conjunction "that" was dropped (ellipted) for easier articulation. I agree with you, though, that "It's time everyone flew" doesn't sound upbeat enough for an advertising pitch, whether or not the times are fraught with the AH1N1 scare.
Other than "It's time everyone flew," what are the other possible grammatically correct constructions of that sentence? Here they are: "It's time for everyone to fly." "It's time to fly." "It's time for everyone to be flying." Are they OK as ad slogans? The first probably, the second middling, and the last a big no-no because, frankly, it creates in my mind images of winged people flying!
So, if you ask me what should be done under the circumstances, I think it's time for those concerned to consider rethinking that slogan for both correct grammar and communication effectiveness.