I have not intervened until now because I have repaired many analogue oscilloscopes of different brands, including Tektronix, but never a 7633.
For me, there are only two possible faults: faulty transformer or faulty voltage doubler.
You worry about things you do not have to worry about: desolder the wire of the voltage doubler that goes to the transformer, whatever the insulation there is on it. You don't have to worry about this insulation !
Power on the scope with the HV board without the voltage doubler and check if you have the - 1475V on the cathode of the tube.
There is no problem to let this auto-oscillator working without load because it is not a flyback and it has a regulation circuit of the -1475V voltage.
If you have the correct -1475V voltage, it is the voltage doubler that is defective, if not, it is the transformer that is dead.
Since there are many connectors on this HV board and that there is a risk of error in reassembling the scope, David Hess's proposal is also a good option.
Again, desolder the wire of the voltage doubler that goes to the transformer.
It seems to me that 40Khz is a bit high, I believe that these auto - oscillators work rather at a frequency close to 25 Khz.
Connect the probe (x10 or x100) of your oscilloscope between the ground and the terminal where you have desoldered the voltage doubler wire and feed the transformer primary as you do with 25Khz, but starting from 0V ... (filament of CRT NOT connected !)
Slowly increase the output voltage of your function generator and check the voltage at the secondary with your oscilloscope.
If you have a very low voltage or nothing, the transformer is faulty.
Nb: some scopes as Philips PM3264 have well known issues of corona discharges, but their HV voltage is 17KV's, not 7KV's as this Tektronix 7633.