I fixed it. On the photo way up there (from 7.3.201

in the bottom left corner are two hybrid "IC"s (small circuit boards with SMD components. The left one (facing left, oriented vertically) seems to be the sync separator/blanking shaper/whateverIdontknow.
Let's number the pins starting from the bottom, which is connected to ground, pin 2 should have the vertical sync pulse output, instead there's some weird oscillating garbage (at least at clean TTL levels).
The SMD transistor on that hybrid "IC" outputting that signal (with you looking at the component side and the pins facing down, it's in the bottom left as well) was bad.
TL;DR
The monitor works for a minute or two when cold. If it's left off for less than a few days, it won't even last two seconds.
Now - freeze spray didn't work. Spraying the area that turned out to have the faulty transistor did little to nothing to the picture. Spraying the area where the yoke connects actually did more, but not much either.
So I started looking at signals and you could see the sync hybrid had some weird needle pulses on nearly all outputs, but most pronounced on pin 2. When you unplug the monitor, you can see the waveform on the scope go from garbage to normal while the signal fades, and when you let it cool down for a day, you could see it start normal, stay normal for a second or two and then go to garbage.
Disconnecting that pin brought back the picture, but no vertical lock, the picture would roll. Connecting the pin to a pull up resistor showed the signal to look normal, but as soon as you connect it, it would go to garbage. Without it, there was no garbage on any of the other pins of the "IC".
I took that "IC" out and looked at what's connected to pin 2 and there's a transistor. Just from looking at it I was pretty sure it's a garden variety NPN and I could see that emitter is ground, collector goes to pin 2 and base connects to a resistor on the board. Just to be sure, I desoldered that transistor, soldered on legs and put it in the component tester and sure enough, what I read is pretty much the same any standard NPN like the BC547 would give, so since I had lots of BC548 and that's pretty much the same, I put in a BC548 and that fixed it. Just to be sure, I scoped the output and it looks fine. It's not getting warm either and I ran it for most of the day with no issues.
That monitor has a crisp and bright image and is capable of a painfully bright image with superb contrast (which required some fiddling with the base and gain pots) and crisp sharpness. I had to turn it down some.