To the various questions...
I haven't gone looking specifically to see if the heater is glowing, but it seems an obvious step. I do get signs of life like HV crackling at power-on and power-off.
@sanleontexas (nice town!), I'll look for that washer. I'd have never guessed it would be a thing. Thanks!
@tunk, those manuals are for an earlier version of the 3279 than the one I have (an S3G-1). I have the equivalent manuals for my terminal, and as @sanleontexas said, it's not much help here.
@tom66, there's no connection for an external monitor, and while some versions of the 3279 could drive an external monitor, this version can't, at least not officially. If I can figure out what signals are what, I might try building up a cable.
@bostonman, yes, I agree that the transistor and flyback are probably OK. I'll have to see if I can figure out what to probe for heater voltage and other signals without getting across some ridiculously high voltage.
I'll try those things and see what else I can learn. Thanks, folks!