And the insulation on the alligator clip is nowhere near good enough to hold off 22 KV, which can jump an inch in dry air.
Approach 2 is more than dangerous, it's sure to kill.
Yes. The alligator clip goes to ground (~earth). Period. But make it _solid_.
The problem with probing is that one's hand can slip. With a TV tube, the stored energy probably won't kill. A 1 mike, 22KV capacitor is quite another thing. There are no second chances, even for cats.
I would wire things up with no HV, then creep the power up with a variac and see how it goes, without touching anything, then disconnect power and walk away for an hour.
Joe Gwinn