If NFL players had sticks I'm sure fights would break out. Stick swinging is much worse than fighting. If you take fighting out of hockey you're going to see a lot of stick work. If you watch European hockey or even College hockey, there's a lot of stick work.
The reason there's no fighting in Olympic Hockey is it's governed by IIHF which has a no fighting rule, but you also have the best plyaers in the world playing in the Olympics so why would they fight?
I play lacrosse which allows fighting, and its much better than swinging for a guys head with a stick.