Don't forget the part about "gives them more performance" which is just an overt lie. They use it because aviation is regulated to high heaven and there are no approved alternatives yet.
Yeah I believe it’s more of an maintenance deal. The leaded gas causes less wear on the cylinders and pistons, so generally makes the plane safer and less risk of a major malfunction during a flight. There aren’t many alternatives with similar characteristics to leaded gas. We used to use it in cars for the same reason, but decided the maintenance/safety/health trade offs were worth banning it.
Not exactly, you can get STC approval to use mogas in most piston engines, it's just a hassle and expense so most people don't do it: https://www.autofuelstc.com/