1. It takes two to fight, and the administration can't know for sure who started it. Trying to determine who "more is at fault" in a conflict has historically NEVER worked, whether we're talking a school setting, an online forum, or a ground war in Asia. Everyone always has an excuse for throwing a punch. So all participants get punished. No confusion, no excuses: if you're in the fight, then you're in trouble.
2. They think that punishing fighting regardless of excuse will be a deterrent, since no excuse will get you off.
3. Even if it's "mock fighting," someone can get mock-hurt on school grounds, then might even decide they "didn't really want to fight" and it's the other guys fault. What a mess. "consensual fighting" works about as well as "consensual sex."
4. Schools are potentially liable if a student gets hurt. They cannot appear to endorse fights at all; they need clear policy set up to designate themselves as non-liable.
5. Other students can get hurt in fights. The school grounds need to feel safe.
Those are some reasons that flew out of my fingers without even really pondering it. Some might be arguable, but I think it's basically stuff like this that results in a "no fighting" policy. There seems to be a big difference between talking about free-form fighting in a school setting vs American violence in general.