Looking at your schematic it appears your HV comes from transformer 601.
Can you locate the HV circuit on your scope? Suspect capacitors have the 2kV rating. It's worth testing each one out-of-circuit with the scope unplugged. Likewise the diodes and resistors.
Weak HV could occur if:
a) a parallel component has become low resistance (diverting current where it ought not), or
b) a series component has become high resistance (inhibiting current).
It's not necessarily easy to measure 2kV directly. Current may be so tiny that it's unable to drive a D'Arsonval meter. Probably same with an inexpensive DMM.
As for the neon bulbs I suppose they indicate whether HV is present. It's hard to figure out what it means when they brighten when you shut off the scope.
Notice the CRT diagram shows the heater wire at the bottom. If it's similar to my Tektronix then that same wire also emits the electron beam. 6.3V combined with high voltage. Anything you try calls for caution so you don't ruin either circuit.
Your aim is to figure out if one or the other voltage isn't normal. Find a wire you can disconnect so you can measure 6.3 VAC safely.
My Tektronix scope developed a problem of arcing in the transformer windings. The trace was a row of ditto marks across the screen. The rapid sparking creating a note singing inside the transformer. I ended up installing a separate small 6.3 V transformer.