The fans all work fine.
I have very little experience working with electronics, and I don't have a voltmeter or anything with me right now, so I've gone ahead and ordered a replacement power supply and logic board. If that doesn't fix the issue, I don't know what will. Even if one of the parts turns out being unnecessary, I guess it couldn't hurt to have spares on hand considering I expect to use this machine for a long time and something else is likely to fail down the road. When they get here I'll see what happens.