It's been a while, but I finally got the money and the time to continue this project.
I replaced the capacitors using a Bob Roberts cap kit and I replaced the blue transistors on the neckboard, and this is what I am left with:

At least the jailbars are gone

The smudge in the upper left is on the plastic, not the monitor. Different inputs cause the monitor to "whine" at different frequencies, but the image looks the same the whole time. It looks that way with no input, too.
Also: adjusting the pots on the neckboard doesn't change the output AT ALL (not brighter, not darker, nothing) and the brightness and contrast controls on the main board don't seem to affect the picture either. The vertical hold pot seems to make a difference and the screen and convergence still affect the image, as well.
I only come back here because you have been so helpful in the past. I'm not sure what else to do short of reflowing all the solder joints on the entire board!
Thanks for your help.