Since there appeared to be nothing physically wrong with the amplifier, I plugged it in and turned it on. No blue smoke. I plugged in the speaker and turned it on again. A "vorp" emanated from the speaker, and fiddling with the knobs caused crackles and pops. I grabbed a bass guitar I had gotten at a flea market and plugged it in...

And sound came out. It actually worked! The pots were undeniably filthy and the input jack intermittent, but ultimately the amp was functional.

The power switch and LED were glued into the chassis, and the wires between the board and the transformer through the hole in the chassis soldered in place, so I unbolted everything I could and pulled it out of the way.

Then I grabbed napkins, q-tips, isopropyl alcohol, and an electrical contact cleaner, and set about scrubbing everything down. When I got to the face of the chassis, though, something strange happened.

Here's what the chassis looked like before:

And here are the input jack and power light after:

It seems that someone sharpied over a green outline and the model info for the amp. With a bit of googling, I was able to find out what amp this is. It's a Quantum 'Terminator' FX-15 amp. And boy has this one been trifled with. Here are the only photos I could find of an original:

Blurry, but undeniably the same amp. It also explains the plethora of extra screws on the front holding the new grill cloth in place.

With the chassis clean, I replaced the thermal paste on the power amp heat sink.

And then put the amp back together and turned it on.

Since once again nothing exploded, I plugged the bass back in and confirmed that audio still came out. Success!

Next I'll set about repairing the speaker, which has a small tear in the cone.