I finished a first-person puncher run of RAGE on Nightmare difficulty a few days ago, where if it was possible to get close enough to punch an enemy, that is how I killed them. If they were unreachable I used wingsticks (a three-pronged bladed boomerang). There were two enemies I had to use a gun on, although if I had better timing it could probably just be one. Interestingly, the final mission requires only punches. I'll have to redo and record this someday.
Now I'm replaying Shadows of the Darned (not actually darned, but I'm guessing the real name would break forum rules) on the hardest difficulty because while it lasts, it is basically the greatest game.