Q2W...you're almost right.
Gas does cool the engine, not oil....no question there.
But, if you're using enough gas for proper jetting (ie rich of stoich, 11:1 or whatever), there will be enough fuel to cool it.
You'll need some huge jets if you're running something stupid heavy like a 10:1 ratio, but as long as the AIR:FUEL ratio stays the same, the fuel:oil ratio won't matter.
If you were to up the oil ratio to 10:1 after jetting your bike to 32:1, then yeah, there won't be enough fuel, it'll run lean, and you'll destroy the piston, because that extra oil changed the air:fuel ratio.
Only so much liquid will pass through a jet, so if you're jetted for pure fuel (for simplicity's sake) and then go with a 1:1 fuel:oil ratio, it will be the same effect as giving the engine half as much fuel.
Change the jet out for something that flows twice as much and you'll be back to the same air:fuel ratio, but with all of that oil mixed in with it.

Something else to consider...when a bike is run on methanol, the air:fuel ratio changes and you're putting in something like 15-20% more fuel to maintain rich of stoich.
That means the engine is seeing 15-20% more oil too, if you kept the same ratio that you used with race gas.
Most of the time though, the oil ratio used by alky 2-strokes is heavy, like a 20:1 ratio.
On top of the heavier ratio, there's also more oil going into the engine because the jets are bigger too.
Those engines are probably seeing the equivalent of a race fuel engine mixed at 15:1, but I'm too lazy to do the exact math on that.