wondering if anyone here is confusing pipe with muffler and calling either "exhaust"
muffler isn't needed, a short piece of header pipe is needed
Same, a header is a particularly good idea on most engines, it's going to direct the exhaust where its supposed to go, and helps keep engine temps more even, if you don't use a header on one of our purely air cooled engines and that exhaust is on the side that gets the blunt of the air cooling then you can easily damage the cylinder through warping by having too much heat escape from one side, or so that's the theory.
Aircraft really don't need mufflers, there's no reason to attempt to quiet the engine, it's not operating close enough to the ground to be considered too loud. Pretty sure though I've seen headers on every piston driven aircraft engine, would they operate badly without them? Who knows, don't know of anyone who takes the exhausts off their airplane motor and kills themselves with carbon monoxide poisoning in their cabin.
I think it may be all up to motor design, if it's a naturally air cooled motor chances are you will want a header of a minimal size installed unless you want to chance warping it. Forced air cooled is a bit less likely to need a good header, but remember in that case the exhaust is probably helping to remove the excess heat from the engine area and leaving just enough that the fan can handle the rest. A water cooled system is probably the least affected by the header or lack of, but again the exhaust will be transporting the excess heat away from the motor, and leaving only what the cooling system is meant to handle, trapping too much heat in or around the surfaces of the engine will overwork the cooling system and all that leads to early failure.
As Steve mentioned, 2 strokes rely pretty heavily on exhaust design, you don't have to use anything spectacular, but not using anything will make it run far worse than having at least some kinda pipe.