The two biggest disciplines, sport climbing and bouldering, are more or less safe so head game is a non issue for the elite. It takes people varying levels of time to convince their brain that it's actually okay to lob off a piece of rock, but most intermediate and above hobbyists that go climbing outdoors once a week are probably not limited by fear.
"Trad climbing", which involves placing the (often dodgy) protection in the rock yourself, is more about maintaining composure in the face of a maiming (or worse) if you fall off rather than strict difficulty of the climb. Your brain definitely has to be wired different for that.
The cream of the crop are generally genetic freaks (in terms of finger strength and/or muscular metabolism) that have been doing it since they were toddlers. But it's also a skill sport, Adam Ondra is clearly the best to ever do it and there are plenty who have better raw strength numbers than him.