Guesses:
1. Maybe D560, D561, D614, or D616 is bad. If, say, D560 is shorted, D565 won't bias to the right point for triggering. If D561 is open, D565 won't see the trigger pulse. If D614 or D616 is bad... well... idk how this part of the circuit works, but these seem like they're part of the trigger inhibit circuit so maybe they'd cause issues.
2. V625 is doing something bad, or R627 has gone out of tolerance. These would cause the tunnel diode to be biased improperly maybe?
Things to check:
1. Maybe D565's quiescent voltage in the state when it should be ready to accept a trigger. It'd be nice to know the quiescent current, too, but I'm not really sure how I'd access that.
2. Just straight pulling V625 and putting it in a tube tester... or swapping it with it's partner. It's just hellish to access, like many things on this incredibly densely packed scope.
3. Maybe I could just pull the "I" cable. Would this change the Q-point in a dangerous way of anything? Would pulling this wire leave D565 in a ready state?
4. I have a P6042 current probe. Feels like I should stick it on the "I" cable on the A side and B side and compare to see how much current V624B is trying to steal/contribute to D565's biasing. It has a 1 mA/div mode. Probably going to be noisy as hell but maybe with enough filtering it'll be ok.
Any other ideas, folks?