Ok, here's what the case and CRT look like today, all nice and restored other than the burn mark. The lighting makes it look a lot more yellow than it really is. It's a very pleasant beige now. The full submersion retrobright seems to have brightened it up just a little bit, which is exactly what I wanted.
I didn't replace all of the caps on the analog board, but I did replace all of the ones
on this upgrade list. I had about $15 left in my "geek budget" for the month so I decided to just replace these specifically at first.
Here's my tape job on the anode cap. I still don't feel like the metal clip is holding on very tight.

I put down a square of tape first (not touching the hole in any way), and then another layer of tape holding the cap down:
Then I put the case back on, flipped the switch aaaaaaaaaaand... Black screen, no bong, and a low "whooooorrrrrrrrrrrr" buzzing sound. It's not high pitched like a mosquito, it's very low. I quickly turned the switch back off. After a few minutes, I was brave enough to try again, and this time it went "whhhhoooooooorrrr, whhhoooooor, whooor, whor whorwhor" and I shut it down because I was terrified.
Anyone want to guess about what is going on? I'm suspecting that the anode clips aren't making good contact, but that is just a guess.