Some sports are just naturally competitive activities though, aren't they? Soccer, basketball, football, hockey, volleyball, handball.. You are trying to beat the other team in some way, usually by scoring more points than them. I don't see a way to make that a non-competitive activity. Am I misunderstanding the objection?
Maybe the problem is the way some coaches, teachers, and schools approach the events, making them more competitive than they need to be? Something can be competitive and fun at the same time, there doesn't need to be a crazy amount of pressure to win. When we had Canadian football in high school gym class, I remember it being more as learning the basics of the sport, how to throw the ball, how to hold it properly, the rules of the game, how to work well as a team, etc. rather than just "win at all costs". But maybe that's just my personal memory of things, since the sport was new to me? Our teacher/coach split us up into 2 groups - those who were better at the sport and those who weren't. I was thrown into the "not as good" camp, which I didn't mind, since I knew nothing about the sport at the time really.. so it made sense. I did not see a crazy amount of competitiveness in the "better" camp of students either, though. They were competing to win, but each time I glanced over to the other pitch, I saw them mainly having fun. I went out to watch a couple Canadian football matches against other high schools as well, and those were definitely more competitive, but it didn't seem negative to me either - i.e. I seemed to see a healthy amount of competitiveness. Is the problem that in some American schools this is taken to the extreme?