This is possibly the best video I've seen so far Dave, awesome! I just love old analog stuff. I shall relate my tiresome tale:
A few years ago, I was gifted an analog scope (trash-found originally from a college here in Toronto, Canada), Philips model PM-3200 (1969 vintage) and I plugged it in and it sort of worked, blurry as hell, unreadable bah.
So, armed only with ohm's law and a few tools I took it apart one day just to see what was up. I used the hfe checker on my multimeter (I know you say they are useless, but I used it as a pass/fail) to test all the transistors inside (which were helpfully mounted in plug sockets). Identified a few dead ones, popped down to the shops, grabbed off the shelf replacements, slapped 'em in and blammo! It works.
Not perfect of course, spent about a week calibrating it and it's still a bit off but for a novice like me it's a godsend, use it all the time!
I especially like your point about safety - there's some narsty voltages inside CRTs and I nearly killed myself but was saved by my insulated multimeter probes which had accidentally contacted a -1500V rail boo! Still have a nice melt mark on the probe to remind be - be really, really careful.