How much do you guys thing this plays a role in a outcome of a fight? E.g if a fighter is facing a KO artist or someone they have been avoiding.
It plays some role, but we the viewers can't quantify it. Some people have had the same mental edge or emotional edge over an opponent and lost, yet we never know about it. The examples we hear (Buster Douglas' mother going into the Tyson fight, etc.) are only when it actually "pays off" in front of our eyes. Had Douglas lost that fight, it wouldn't have been seen as an advantage but instead a distraction. But since he won, history remembers the unknown in that way.
Look at Bute vs Pascal. Bute had no confidence in his own abilities after the Froch fight, and it showed. Only in the last couple of rounds Bute finally opened up, but too little too late. And certainly fear can cause someone to completely freeze up.