The voltages on Q4 show that my interpretation of Q4 on/off is reversed.
These old HP schematics where transistors are drawn both "right side up" and "up side down" always make my head hurt.
The most common convention is to place the transistor terminal, emitter of PNP and collector of NPN at the top where positive voltage is also at the top. Negative voltages and transistor terminals connected to negative voltages are drawn lower. Here is an example where PNP transistors are drawn two different ways in the same schematic. Crazy. This just makes it harder to understand the circuit.
After drawing this circuit on paper the "proper" way, it is obvious at a glance how Q4 works. When the scope is off, the -15 Volt supply is at zero. C18 discharges through R44. When switch S1 is in "normal" and the scope is first turned on, C18 looks like a short and holds Q4 off. R44 charges C18 with current toward -15 Volts. When the voltage on C18 exceeds VR4 plus the base-emitter drop of Q4, about -11.2 Volts, Q4 turns on.
When "deep erase" connects R45 to ground through switch S1, C18 is discharged and Q4 is turned off.
In "normal" with Q4 turned on, current through R45 turns on the LED in U2 and the output transistor of U2 also turns on. This action turns on Q5 and shorts VR3.
VR3 causes the CRT cathode to be 100 Volts more positive than the grid, the CRT is more deeply cutoff. This protects the CRT from burning when the scope is turned on and turned off, or when electric power is lost momentarily and comes back in very quickly.
After a short delay by C18 and R44, the protection is released.
Based on all the voltage measurements, I think the problem is the LED inside U2 has failed open circuit. The output transistor inside U2 cannot turn on and the extra 100 Volts keeps the CRT deeply cutoff.
The single most valuable measurement is the voltage drop across R45 (270 Ohms). Using a 4N25 datasheet as a reference, 10 mA of LED forward current is used to describe the transfer characteristics. Data found for U2 part number 1990-0607 is too minimal to be 100% conclusive, but 625 uA seems not sufficient to turn on U2.
4N25 is not suitable for replacing U2. A good parametric search is necessary to ensure a replacement capable of withstanding 120 volts across the emitter-collector of the output and 4 kV across the input side to the output side.
You indicated that U2 is in a socket, do you know how to test U2 out of the circuit?