Another great video Dave. You are a natural!
I suspect there's one root cause for all the self-test failures. With the ADC clearly not working, the analog section cannot take any measurements. Thus any tests requiring the ADC to work must fail. Since the self-test is complaining about the ADC FPGA, that's the place to start. I don't think that you pulled that board in the video, but if it's another Actel ACT 1 FPGA in a socket, the first step is resocketing that FPGA after using a bit of contact cleaner with the hope that a bit of corrosion is the culprit.
The good news is that the entire digital section seems operational with perhaps the exception of the missing I2C peripheral.
The 39.xxxx MHz crystal oscillator in this instrument is a wonder. It takes up a huge amount of board space! I would have guessed that with all the attention and space given it, this oscillator was supposed to be some sort of ultra-stable clock source for all of the spectral measurements this DSA is supposed to make, but then I would have expected an oven-controlled crystal if stability was that important. I would generally cheat in this part of a design and just use a canned crystal oscillator in a metal can with a 14-pin DIP footprint. It would have been a lot smaller and my designs generally were crimped on board space. Also, it wouldn't need -12V. You could certainly afford one of those in a $20K instrument.
Strange that something killed the primary section of the "new" or "repaired" power supply. A spike that got through and also killed the negative voltage regulator? I can't tell from the video, but it looks like HP bought that supply from a power supply vendor instead of developing one on its own.