It's true that you hear the term "cocktail" a lot less often than you hear the name of a specific cocktail. In fact, "cocktail" has a kind of old-fashioned sound to me (no pun intended, but there is, in fact, a specific cocktail called an "old-fashioned"). I'd say the term "mixed drink" is more common. "Cocktails" is likely to be seen as a neon sign in the window of a bar that (1) is old or old-fashioned, and (2) is (or was when the sign was new) trying to attract a more upscale clientele than those who merely drink beer.

Also, liquor laws vary a lot from state to state, and in some states there are two (or more) classes of licenses. There could be one that allows the sale of beer only, one that allows beer and wine only, one that allows beer and wine but only with a meal; and one that allows the sale of beer, wine, and distilled liquor. So a bar that displays a sign saying "cocktails" might be trying to inform you that they're NOT just a "beer bar." Alternatively, the sign could say "liquor," and I think that, in fact, is more common.

Which reminds me, I was once highly pleased when my British host asked, "Would you like a drink?"—and then disappointed when I realized he was proposing to make a pot of tea. In America "drink" used that way usually implies alcohol, if not liquor. I don't know why, but "Would you like a drink?" has an entirely different connotation than "Would you like something to drink?"