The board was intermittently dead, which is the worst kind of dead. Sometimes it powered on, sometimes it didn't, and tapping it would change its mind, which meant it wasn't a clean failure I could measure. A component that's properly broken is almost relaxing by comparison: you find it, you replace it, you move on. An intermittent fault that responds to a gentle prod is a connection problem, and connection problems hide in the boring places.
I'd convinced myself something on the board had failed, some chip or capacitor that I'd have to identify, source and replace. So before ordering anything I sat it under the magnifier lamp and just looked, which I should have done first. There it was: one joint near the power connector with that dull, grey, slightly cracked look. Not the bright concave fillet a good joint has. A cold joint, or one that had worked itself loose under years of thermal cycling, holding on by a thread of solder that lost contact whenever the board flexed.
No new parts. No schematic. A dab of flux, the iron held on the pad until the old solder went bright and flowed properly, a touch of fresh solder, let it cool undisturbed. Powered it up and it stayed up, through every tap and flex I could give it.
The lesson I keep relearning at this bench: suspect the joints before the silicon. Solder fatigues, especially around connectors and anything that gets warm or takes mechanical stress. A reflow costs nothing and a minute, and more often than I'd expect it's the whole repair. Look before you buy.