Old drifted carbon comp high value resistors. 
Might as well replace the 4700pF caps too....
The resistors are all 1% metal film. Even the logic pull up resistors are 1%. The only engineering in this instrument is over engineering. It is almost like the Scottish design team were trying to show the US HP office how good the Scottish office were, rather than building a commercially viable test instrument.
The big HV resistors look ceramic types.
Today I washed the HV board with IPA to clean away any contaminants that might lead to arcing. That made no difference.
I am leaning towards the fault being the wires/connections between the crt deflection plates and the Y deflection amplifier. Bad connections would be consistent with the step changes in Y gain I am seeing. Maybe a floating deflection electrode in the CRT would explain the interaction with intensity. Right now it is the only thing that fits the facts.
I ran the HP 3721A for about 6 hours today to attempt flush out the fault. The 3721A gets HOT HOT HOT. I measured 51 deg C on the metal near the power transistors. Even with the ugly fan, the rear half of the 3721 was uncomfortably warm. I am going to fit TO-3 heat sinks to try and dump the heat into the air and not into the chassis. I am also going to replace the super ugly external fan with an internal 80 x 15 12V fan and make a nice bezel with a period correct grill to cover the hole through the side of the enclosure.
I tried to video the HP 3721A fault seen on the crt, but I got the exposure completely wrong.