The Super Bowl is the only "Big Game" they refer to in the ads. Sure there are other "big games", but these are Super Bowl ads pure and simple.
Why make an ad you can only use once, when you could easily tweak it and be able to reuse it? Like you said, its not like anyone won't know you mean the Super Bowl, when you're talking about the big game on Super Bowl weekend.
I could! Unless I did something ill-advised, like saying "Super Bowl", which in fact does only refer to one game.