Bars are exempted because bars spend a lot of money on their TV hookups, more than regular people would for a similar setup, and a good chunk of that money goes to the leagues. Even then they do have some limits on whether or not they can advertise themselves as a place to watch the Super Bowl (or the NCAA Tournament which is also rigorously defended). If you’ll notice most sports bar advertising generally talks about “the big game”, which works for two reason, it means they don’t have to change the commercial as often, and it keeps them from getting in any trouble with the leagues. You to can charge for a Big Game party, assuming you don’t violate zoning regs of course.
I’ve been reminded of another reason: in many markets, bar-owners own lots of season tickets. Back in the olden days, the bars couldn’t show the games unless it was sold out (still the case, but these days, most games sell out, right?). So the bars would buy up the season tickets so their customers could watch the game.