The point system might declare winners in a silly way, but why care about the points?

I would expect that one reason we gravitate toward competitive games, especially those with points, is that we can't help but treat them as signals of status. I'd in fact expect that if you ran a psych experiment with subjects playing a game, and the experimenters instructed them to play in one fashion but meaningless points were awarded under a different criterion, that people would be dramatically swayed by the point system (and possibly unaware of this fact).