Okay, had lunch and the little one is napping, which decreases the risk of someone wanting to lick the HV part of the PCB back to background levels.
As a summary of where I was: I traced the issue down to the horizontal HOT getting driven for a bit on startup/shutdown, but something was shutting the driver chip down immediately after startup.
Last time, I thought I saw the signal into the HOT shutting down *before* the driver signal did. I think this was a glitch of how I measured and how the chip works: the chip doesn't shut down the oscillation, rather it kills the power to the output, making it taper down a bit before stopping. During the tapered down phase, there's not enough power to drive the transistor connected to it so it looks like the HOT stops oscillating before the chip does. Here's a zoom-in:
