The "green stuff" is a lubricant and insulator that keeps any wear and possible arcing that might occur on the pot's wiper to a bare minimum. It also provides some mechanical resistance so the pot feels "smooth" when adjusting it.
I would use a high quality silicone grease available today. You want something that's more of a paste. The intensity control is in a 1400V circuit, with the focus control next in line. See the schematic "CRT Circuit" in the link I posted.
Since you've narrowed the noise to the power supply, it's probably time for recapping the electrolytics!
You can't depend on value checks alone, so get your ESR meter out an have at it.
Is the battery pack connected? If it is and it's bad/dead/shorted, that could be causing the power supply to squeal too.
... Good thing since I don't know know how to fix a failing EHT transformer
There's usually no "fix" to those transformers. However, fixing the rectifier/multiplier can be quite easy. Just HV caps & diodes.
T