Some engines are also substantially cooled by oil, but those are either older designs (think “air-cooled” Porche) or industrial prime-movers.

Most piston aircraft engines are still air-cooled which really means air and oil cooled. The oil is a big part of getting heat out of various parts of the engine.

That also makes them harder on oil as the piston/rings have larger tolerances so they don't expand and bind up during operation. That means greater blow-by at startup and when operating at lower temps which puts a lot more combustion byproducts into the oil. Ultimately you want to run an aircraft engine in the upper part of its range (65% power) continuously and don't let it get too cold.

This is also true because 100LL still contains lead and at lower temps the lead combustion byproducts precipitate out of solution, coating everything in metallic lead, lead oxides, and various other lead compounds all of which are really bad for engines. Converting to unleaded nearly doubled engine life in autos.

Many modern engines have valve rotators and hydraulic lifters. Oil pressure is fed to a lifter that sits between the valves and the cam and automatically takes up for any variation in the system, ensuring valves operate correctly. If you ever wondered why car engines don't need to have their valves adjusted every 20k miles anymore - that's why. In some engines if these leak down after shutdown it can cause trouble starting because the valve timing will be off until oil pressure re-fills the lifter.

Rotators are little spring mechanisms that compress and when uncompressing try to rotate the valve in one direction. This causes the valves to rotate a tiny bit with each cycle. Often there are hot spots and exhaust valves especially often have no good way to shed heat yet are exposed to extremely high temps - so they shed heat when they close and are in contact with the head. If they don't rotate the slightly hotter spots will continuously build up heat eventually destroying the valve. The rotator keeps that from happening. (Some engines use sodium filled valves to help transport heat away from the valve face).

I always found it surprising how tiny variations in wear or even a few degrees of excess heat can end up destroying an engine.

You are quite right, the oil also serves to transport the heat away.

Some engines, especially high mileage engines from Detroit, are also substantially fueled by oil. In a sense.