As for what is simple or not, just check out the bridge fight in Operation Annihilate. All he really would have been required to do in the long shots would be to help restrain Spock with a few other guys. Pushing him down, really. This is the one bit that actually causes me to wonder, because it's ridiculously simple.
I did check out that fight scene, and it was rather more intense than what you describe. "Kirk" leaps onto "Spock" from several feet away and gets tossed into the bridge railing, catches himself on it, recovers his footing on the stairs, and then grabs Spock from behind along with the rest. That's a maneuver where something could easily go wrong -- the performer could hit the rail or the step wrong and injure himself. Not to mention that this is in the tight confines of the bridge set where there are lots of other hard surfaces and sharp edges to worry about. Not to mention that the light bulbs underneath all those console buttons were really hot. There are a lot of ways that set could damage a person if a stunt went wrong and he landed in the wrong place -- not to mention the risk that the person could damage the set.
It's one thing to have Shatner do a fight scene in a nice, open corridor where there's nothing but walls and a floor. But on the bridge set, there is so much more that could go wrong, and I can absolutely understand why they'd leave such a stunt to the experts.
Another factor is the number of people involved. A one-on-one fight is one thing, but when you've got multiple people in a melee, there's a lot more that could go wrong if your performers aren't perfectly in sync. In this case, there are four stunt performers directly involved in the scuffle, and that's exponentially more complicated than a scene with two performers.