So after spending last weekend looking over things, I put everything back EXCEPT the finals and the speaker. I rigged a little power cord with alligator clips (I don't have another plug for this radio) and a circuit breaker on the hot line. I connected the hot to the fuse in the radio and the ground to the radio chassis and powered up the supply. I was greeted with the radio lighting up 21.2027 MHz. I couldn't hear anything, as I had the speaker removed.
So the next thing I did was resolder the speaker and the the finals. After finding a nylon washer for the internal hot wire in the finals, I did the minimum replacement of screws and powered the unit back up. It immediately popped the circuit breaker.
Guessing that the finals were bad, I removed the finals and powered the rig back up. The display came back up, but no sound out of the speaker. I started adjusting the volume and RF gain trying to hear something. I then noticed that the mode switch was set to "LOCK", so the transmitter was on. One thing I did notice is that no power was coming out - I had the internal antenna connection connected to my little QRP dummy load that has a rectifier output and I had my multimeter connected to it and there was no DC voltage coming through the dummy load and no read on the frequency counter in my antenna analyzer.
While there is a slight flaw in my troubleshooting, I'm fairly confident that the problem is the finals and I will be taking a closer look at the module tomorrow.
Incidentally, I looked up the price for new finals at RF Parts. $60 for a pair, and $25 for one. Of that's the problem, it'll be a fairly inexpensive fix.