It's pretty limited. Avgas doesn't mean jet fuel, as used by commercial aviation, it means the stuff you pump into piston engine light aircraft. Plus, at the risk of being obvious, the exhaust is either emitted way up in the air or, if at ground level, out in the middle of nowhere at an airport.

Yes, there are light aircraft that run on diesel.

Not a pilot or aircraft owner, but my understanding is that part of the problem is because certifying a new aero engine is insanely onerous (as is certifying anything that goes on an airplane), a lot of basically obsolete designs requiring leaded fuel were produced long after auto engines switched to not requiring leaded gas.

Why would an engine require leaded gas, you ask? Because it helps keep the valve seats from degrading. We have other anti-knock additives, but nobody has figured out a replacement for the other properties that lead has.

And since aircraft are usually extremely long-lived, we'll be stuck with this problem until a substitute is found. Merely certifying new engines that don't require lead won't solve the problem because of the remainder of the fleet that does require leaded gas.