The actual act of fighting in hockey rarely leads to brain injuries. Hockey fights are very sloppy, most fights last less than 30 seconds and because they're grabbing each other's shoulders, more often than not, not a single punch lands cleanly. Slipping and falling, pushing each other to ice, contact with the ground usually is what causes brain injuries. Even in the given example. So if the idea is to combat brain injury, taking fighting away doesn't work to that end basically at all.
I'd venture that the majority of concussions in hockey aren't from fighting or even checking but incidental collisions and/or trip and falls. I'm a big MMA/wrestling fan. I've long been very critical of boxing for the reason that I think it's a practically useless sport that promotes brain injury. Hockey fights are not "bare-knuckle boxing", they are two guys clinching until one of them is able to drag the other down. There is some inherent risk in taking someone down on ice but incidents like the aforementioned are freak accidents. And hockey is, inevitably, full of those.