I think it's easier to first test the usual suspects - C624 10uF, rectifier diodes, the other electrolytic caps C611, C612, those big can caps C652, C654 etc. C620 tunes it I believe.
I would not run it with all loads disconnected so you don't make a Tesla coil here.
The 6BL8, I think it lowers the 6DQ6A screen voltage to lower its gain when rail voltages are where they should be.
I have to wonder how that section works- if V634A needs to be turned off (negative bias to give the 6DQ6A some screen voltage) Hmmm. It must use grid-leak, with the high values for R630, R631? Check them and C630 for high leakage.
Does that make sense? If the regulator V634 hold down the 6DQ6 it could not start up.
Something else I would try is with power off, use a muscular sig gen and inject say 30kHz several VAC at the secondary winding for filament power (xfmr at D672 and GND). You can pull the 4 tubes to off-load that winding. Just to see if there is an obvious heavy load on one output or the primary, and you can poke around with a scope or voltmeter. Not to take it to full voltage but look at at even 1/10.
If you are suspecting the primary has shorted turns, then I'd use a Ring Tester.