Maybe I'm just missing something that can be explained to me... maybe not...
But adding more oil ain't gonna' reduce internal engine temperature enough to matter... and it can even increase it (depending). The oil is there for lubrication, the engine parts need only enough lubrication... anything more than enough is wasted. That piston, rod and crank in your picture look plenty "wet"... there appears to be plenty enough oil.
Running a 2-stroke lean will cause it to get hot... a whole friggin' bunch more hot. If I pulled that engine apart and saw that piston my first thought would be "lean", not oil starvation. I'd be thinking lean from either an improper tune or ethanol blended fuel (if I could see the top of the piston I might favor one cause over the other). Typically, in my experience anyway, adding more oil means opening the high side carb screw a little to make them run right... meaning the extra oil forces you to run the engine richer (more fuel), which is what's gonna' cool internal temperatures, not the oil... meaning ya' likely could have just run the thing a touch "fatter" in the first place and not added the extra oil.
Like I said, maybe I'm missing something... but...
When the gas/oil/air mixture enters the crankcase the gas vaporizes, the oil don't. Now you have tiny droplets of oil in there... like a fog, which collects and builds up on the internal parts. Moving parts, such as the crank, sling off any extra, throwing even more of it on places like the cylinder walls and such. What little oil that does find it's way into the combustion chamber doesn't do much for lubrication, it's either burned or just blown out the exhaust (4 stroke engines don't have any oil introduced into the combustion chamber and they survive). It's the vaporizing gasoline that carries away most of the heat (like your evaporating sweat)... running the engine lean means a lot less heat is carried away.
Extra oil can't compensate for a lean tune... it don't work that way.
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