So I swiped one from a strobe kit I had built many years back that had a busted flashtube anyway and installed that one in the flashead. No dice. I figured perhaps that transformer was a dud too, so to test it I grabbed a spare flashtube from my parts supply and hooked it up, and it flashed just fine. I re-installed the original trigger transformer and it still worked. Just to see, I tried it on high power, which worked, but instantly destroyed the little brand-new flashtube I was testing with. Ooops.
I took it home and tried firing it up, zapping it with a hand-held tesla coil to give it a more persuasive trigger. That made it flash, but the end of the tube quickly blackened. Obviously the tube just didn't want to flash any more. And, being quite an old unit, a replacement flashtube wasn't going to be obtainable.
I tried my usual scatter-bid on a bunch of cheap broken ring flashes on eBay, figuring I'd probably score at least one with a usable flashead or at least a flashtube I could salvage. My cheapo bids won on a couple of them.
They showed up today.
They both work perfectly.