Enormous amounts of CRT TVs and monitors had their high voltage sections repaired without ever using an HV probe.
True. We used to just stick the tip of a long, well insulated screwdriver under the rubber anode cap and see if it drew an arc. If it did we figured that the HV circuit was probably ok. But I hesitate to recommend that to someone else because their circuit could be different and doing that may damage it.
Also on TVs since the early 1960s, the HV tripler and HV transformer and most of the HV section was one big sealed unit and it wasn't repairable so if the Horizontal Output vacuum tube (or transistor) was good and we weren't getting any HV then we just replaced the entire HV output unit. FYI In transistorized sets when the HV section failed, it almost always took out the transistor too.
If I was the OP, I'd look at the circuit and see if it has some kind of controller (probably an IC) in it or if it's just an AC transformer driving a tripler circuit and has an unregulated output. Also if the components of the tripler are accessible of if they're in a sealed unit. Also look and see if there is any kind of accessible driver (a large transistor or possibly an IC).
I've never looked at the circuit in an electrostatic air cleaner but I suspect that they use an unregulated circuit since the output voltage doesn't need to be controlled and an unregulated output is cheaper to build. I also suspect the tripler section, and probably most most of the HV unit, is going to be a sealed brick.
Pictures would help.