Mitchell: "I'll just keep getting stronger. You know that, don't you?"
Yup, he did know that. There was no foreseeable limit to Mitchell's growth and power. Mitchell had to die or else grow so powerful, anything would be possible for him. Even getting off the planet. Either Kirk would have to do the deed, or his last order would (hopefully) take care of it.
The real question should be, "why didn't Kirk just grab the phaser rifle when Gary was normalized and blow him away instead of having a brawl?" I mean, aside from "the episode had to end with a fistfight in a ditch."
On-screen evidence suggests that Micthell flung Kirk's phaser rifle too far away for Kirk to recover it promptly (the parabolic arc looked very long and shallow, we never heard it hit the ground). Coupled with Dehner's warning to Kirk after Mitchell and Dehner's struggle ("Hurry! You haven't much time!") and Kirk probably thought that his only option was to try to kill Mitchell with his bare hands. Or, perhaps he was simply trying to incapacitate him until another solution could be devised.
Either way, the fistfight never felt forced or contrived to me. In fact, unforced and uncontrived fistfights are my favorite kind.