The symptoms were that the radio lit up and the display worked fine, but you could only hear extremely faint sound even if you cranked it up all the way. Using my diagnostic powers, I figured the problem was probably in the amplifier, which would probably be near the the back of the radio. Looking more closely, I noticed a capacitor oozing hideous brown gunk. Using my diagnostic powers again, I decided that this might be the problem. (It turns out that leaking electrolytic capacitors is a common problem, known as the capacitor plague.)
I unsoldered the capacitor and removed the gunk as best I could. At least it wasn't as disgusting as the nest of ants that caused my previous electronic problem. The really annoying thing with the repair was the radio's circuit board had big globs of sticky heat sink compound exactly where I grabbed the board every time I picked it up. You can see the white patches in the lower left of the first picture. There was originally much more compound, but after getting it on my fingers twenty times, there wasn't much left. I should have learned to be more careful, but no such luck...
I figured it would be easy to get a replacement capacitor, so I checked the parts supplier Digi-Key. The good news was they had the exact capacitor listed. The bad news is they didn't have it in stock, and it would take 6 months to get it from the factory. Checking other parts catalogs, I found that this capacitor wasn't the easy-to-find commodity part I had expected, but a special short-and-wide capacitor designed to fit into the tight space, that nobody carried in stock. Too impatient to wait for 6 month delivery, I got a standard capacitor, which was the wrong size to mount nicely. I'll get no points for style, but I did manage to wedge it in place by putting it at a crazy angle. (I also put in new heat sink compound to replace the compound that I got all over my fingers.)
After replacing the radio, the radio wouldn't do anything because it needed the security code. Through surprising foresight, I actually had the code, and after putting the code in I found that the radio just gave me static. Oops, I forgot to connect the antennas to the radio. That was easy to fix, since I conveniently noticed before tightening up the security fasteners. With all the wires in place, I tried again and the radio seems to work as well as ever. It's always nice when one of my crazy projects actually works.