My buddy is going to be so happy, no more problem in his methanol burning tractor!
They idle the engine until it about 80c, after a pull the engine will be down about 30c, all while making about 2200hp.

That oil in a large engine is critical when it comes to remove heat is true, pistons and valve springs needs to be cooled.
(After +1000 engines in dyno runs I know)
Now saying that a Top Fuel engine is an oil cooled engine... well, lets look at the numbers.
They say they burn at least 11.2 gallon/second, that's 544.75944 Mj / sec, or 544759,44Kw / sec.
The highest recorded hp in a Top Fuel engine is above 10.000hp, or 7456,9987Kw / sec.
That means they have about 537302 Kw in losses / sec.
1kg of water that's 10c in temperature would boil and evaporate in 0,0049 seconds from the energy loss / sec.
Oil would do better, maybe four times as well(?), but still...
Top Fuel engines rely on cooling from the fuel.

Read the section about fuel; https://en.wikipedia.org/wiki/Top_Fuel
You will also see there that lambda 1 is 1.7:1 for nitro.

Now our glow engines, well they have the oil in the fuel and as such it's subjected to the combustion process.
It would be strange if it didn't get hot, and the more heat it absorbs the less power the engine will make therefore.
If the oil absorbs heat in the crank case of an two stroke before combustion it can decrease heat, but tests would be needed to get any useful information about that.
If you open the back plate of a two stroke after running it you will see oil that has separated from the fuel, if you don't see this, change fuel!

The debate about oil is about as accurate as the rest...