Halfbaked, you've never had the pleasure of flying an overloaded, poorly cowled piston A/C on a hot day, have you? It's great fun trying to clear the mountain without frying the engines.
Student, I think the question and answers are poorly written; ie, how rich or lean are they talking about? I think they are assuming that the "standard" mixture setting is that for best power. Ergo, if the air flow isn't enough to keep the engine cool, then you can increase the mixture a bit and the extra fuel will provide some cooling. Your thought process re fuel required per air density is correct; even running a little rich for extra cooling, you will still not be anywhere near the fuel flow required for the same throttle setting on a cold day. If you set up your fuel system on a cold day, and then try to fly on a very hot day, your engine might not run at all with the mixture full rich, even at full throttle.