So when I got home, fizzygeek started emptying it out. I stopped her and said it was probably worth doing a little testing to see if we could find out what was wrong, and if it could be patched enough to work until tomorrow.
A little investigation revealed that the compressor wasn't running. I decided this was either the compressor itself, the motor start capacitor, the compressor relay, or the control circuit. Accordingly, I hotwired the compressor relay. I figured I'd get either nothing at all (dead relay), a snap from the relay but nothing else (dead compressor or start capacitor) or the compressor would run (broken control circuit or wiring). Sure enough, the compressor sprang to life! So I left it running that way to cool down the contents (it would ignore the thermostat and try to max cool everything, but it should be fine for an hour or two starting from a warm fridge). Then I proceeded to do some research. The control board turns out to be a $91 part, and is famous for failing. There is a chance the problem is with the thermostat or other sensors, but the main board looks like the likely culprit.
So I yanked the main board and eyeballed it good. Sure enough, a bad solder joint on the compressor relay had burned through, as well as a 2Ω, 1 watt resistor looked a little cooked. Both of these failures are common to this board. I fired up the soldering iron and resoldered the relay with plenty of solder ("the bigger the blob, the better the job!"). I didn't have any 2Ω resistors, so I made one up of series-parallel combinations of 1Ω resistors (of which I had plenty), for a total dissipation of 2 watts.
I buttoned everything back up, pulled out the hotwire for the compressor and applied power. Voila, she runs! We lost a little ice cream, some time, and some emotional energy, but we're back on track with refrigerated goods!