I must have been very lucky because a few years ago I removed the Dallas chips from my TDS784C, dumped them and installed low profile sockets (video trigger board won't fit with standard sockets) and the data remained intact. As far as I know there is nothing super critical stored in those though, the SPC constants are but you should run SPC periodically anyway. I think the important calibration stuff is in EEPROM on the acquisition board.
Regarding patching up flybacks, I have only one data point but when I was a teenager I found a 25" Sharp TV by the dumpster at the apartment complex where we lived. I dragged it home and found a bad capacitor next to the flyback and then the flyback was arcing to the frame. I glopped a bunch of hot melt glue over the crack and it worked fine for years, it was still working when I gave away the TV several years later.
Looking at that photo of the screen again, the fact that the image appears very bright and somewhat washed out also suggests the EHT has gone way high, I'm not familiar with how it is regulated in these monitors, thankfully I've never had to try to fix mine. Given the unusual NuColor display runs at a high field rate I'm wondering if the HV is separate circuit independent of the horizontal scan? If that's the case it might not be too hard to adapt it to use some other flyback transformer. I do love those NuColor displays, it's such a fascinating and unusual technology and it produces a very unique look, a sharp clear image as produced by a monochrome CRT with no screendoor effect of a shadow mask, yet in full color. There was only one other application I'm aware of where it was ever used, in a small studio broadcast monitor, I'd love to find one of those but they're quite rare.