Smoke test with new power supply, and opal's motherboard in diamond's case: no go. The fans spin up and then spin down immediately. On roughly one boot attempt in four, seemingly random iid, it gets as far as the single beep that is normal, and lasts a few seconds before shutting down. But nothing ever comes up on the video output. Same behaviour if I disconnect all the drives from the motherboard, except that (if left connected to the power supply) the drives by themselves will remain spinning. So it's pretty clear that the motherboard (or something attached to it so intimately as to effectively be part of it, such as the CPU or RAM modules) is fatally damaged.
Conclusion: This motherboard is dead. I have to buy a new computer. If I'm lucky, I'll be able to salvage the data from at least one of the hard drives. All the important stuff is on both, via RAID-1, so recovering one is enough to salvage that; and all the critically important stuff is off-site, so it wouldn't be an absolute disaster even if I lost both drives; but some low-priority (replaceable with annoyance) data is striped on RAID-0, so recovering it requires recovering both; and obviously it'd be nice to salvage as much as I can.