I brought the power supply home and replaced the old one, though, and the computer is still pretending to be a doornail. I'm getting a little more than I had before (a twitch from the fans, where there was nothing before), but it's still a far cry from a successful boot. Looks like the next thing to try is switching out the motherboard. I have a spare one (that I'm lucky I haven't sold yet), but it's a Pentium 4 board where the old one was for an Athlon.
Does anyone near me (you know who you are) know where I can get a P4 chip cheaply, without waiting for a special order? I went looking before for one (when I got the mismatched motherboard and CPU at the computer sale here a few months ago) and the best option was to order one in.