Your entire second paragraph discusses the role that it plays in the game, but does not actually do anything to describe it as fundamental to hockey. There used to be no forward pass a hundred years ago, and I imagine most of us remember two-line offsides. Getting rid of fighting would absolutely change the NHL, yes. But it's still just a rule like any of the others. And while I understand the difference between tournaments and leagues, I'm making a point about the role of fighting in hockey. Tournaments are absolutely relevant examples.
I've played hockey my entire life. From the moment I could stand and hold a stick until my beer league game later today, I've played this game. I'm just a beer league duster, but I have never once played in a league that allows fighting. It may not be pretty what we're doing, but we're still playing hockey. If you take away the fighting, it's still hockey.
I understand the role that fighting plays in the NHL and other professional/junior leagues. As a part of those leagues, it directly effects the game. But if 99% of the people who will ever play this wonderful game don't fight, I don't see how anyone can argue that it's actually intrinsic to the game. It's cool if you like it and think it plays an important role in the NHL. But it isn't some sacred act that must be protected at all costs.