So... tonight I asked myself, "Self? Should we restring the guitar, or work on the Joust machine?"
"Well, self, you have a good idea on what to do on the Joust machine."
"Not really, self, but we will try anyway."
I then broke out the soldering tools and went to work. The first thing I did was connect W2 and W3 jumpers on the power supply board.
After doing that, I looked under the board and pin 12 on where the board connects to the isolation transformer was suspect. So I reflowed it and pressed it gently back towards the board.
I turned on the machine, and got lights!

I still wasn't getting anything from the ROM board, but when I tested the sound board, it was so loud that I thought I was going to jump through the roof. I took a chance an pressed down on the interboard cable. Then I got this:
Success! I moved the yoke until I was satisfied with how straight it was. There were still some RAM errors so I fiddled around some more and got it stable. I tried to go into settings, but odd things were happening in the menus. I finally figured out that the second player joystick is permanently pushing left. I will have to check that out later. As you can also see above, I have some degaussing to do. I tried to fix it, but I will probably need to move the machine before doing final degaussing.
I was also getting a little curl at the top of the display. Convergence could be better, but I don't think the rings I have will move without disintegrating.