I think a lot of the American love for the school spirit deal in HS and college, as well as the public team worship for pro sports, is just a method of making the motto of the US: E pluribus unum, a reality. It's a way to unite disparate populations and give them something in common. Yeah, you're __________ (Black, White, Hispanic, Asian, Mutt, none of the above), but as long as you are in Giants colors, and I'm in Giant's colors, then that no longer matters (or its not supposed to) because we are both sharing something together - in the Giant's case usually heartache.
But the unity can be and is a very real thing. Ask any sports fan in a winning city or campus what's it like, it can be a very overwhelming feeling, one that does overwhelm a lot of other differences.
And, as American as it is, it's not omnipresent. Baseball does not have cheerleaders. And since soccer was imported without cheerleaders it continues to be sans them. Track and field, motor sports, horse racing, swimming and a host of others do not have them either. Pretty much it's football, basketball and wrestling.
And in football its a side-show, its just that the rah-rah nature of cheer-leading went perfectly with the rah-rah nature of the game and the spectating sides were split on opposite sides making it pretty easy to work the cheer deal. That and it had its real start in college which is all about cheers for the above reason. But since football is outdoors, it's big and the crowd is far away, and in those helmets and all that noise is not really a factor in the game. But in some things like college basketball it makes a huge difference.
The clip below is of KU singing it's alma mater song and then doing the 'Rock Chalk Jayhawk' chant before the game. They do this for every home game. Look at the amount of KU blue in that fieldhouse, and listen to all of them using that stuff to all get on the same page, the cheerleaders can then use that unity to really rock the house. Imagine how loud it is in there - all directed down to the court itself - during the game. Last year against Mizzou they were up to 120db inside that field house, that's like trying to play b-ball inside a couple of jet engines.