I am highlighting a limitation. Also, I never saw them break up fights by being able to "carry" or restrain players away from each other like other sports. It's much harder to do on the ice. A lot of times, the linesmen are mostly separating the players by strategically intercepting between the players' line of sight by pushing in between the players. On top of that, the players have enough sense that they should not hurt an official, especially when it's not advantageous back when it was the meta to have brawlers. However, there were moments when the players ignored the linesmen interception and kept going. Small example: https://m.youtube.com/watch?v=yDkNvuuENwo

Also, I'm not saying the league wants more fights. They want to control those fights because they are profitable for entertainment, IE the fans and surrounding community are entertained by the drama. Having a fight break out for no reason other than to cause harm was not fun or much for entertainment. However, talking smack or telling someone off for bad plays/manners is fun to see.