"Audio Modulated". If the HV is working I would try to bypass modulation circuit. I suspect the audio transformer is toast. With power off and caps discharged, read resistance across secondary of "T1". I'd bet it's open.
From the pictures of that pc board, I would guess that is all audio modulation. There are no multiplier components that would indicate it is part of main power. That suggests that the black box under the tube is a self contained HeNe supply. The HV output would feed anode through the secondary of T1. It could likely be bypassed to get the tube running.
This is all speculation. If I had it in my hands I could probably figure it out, but trying to troubleshoot from pictures is challenging. If you decide to tinker with it, remember that HV is no joke. Although a HeNe supply probably won't kill you, it will make you choke on your chewing gum.
EDIT:
I did some searching and found that the modulated HeNe student laser sell for $670.00 new. That PC board is in fact an audio amp circuit. Just plug in microphone and transmit. Receiver sells for 150.00. But you can easily build one by connecting solar cell, photodiode, or non-diffuse LED to an amplified speaker ( like on computers).