OK... this is weird. As suggested by
@Ericloewe, I added a new user to IPMI, just to see what might happen. At first, nothing much interesting - the new user took, and I was able to power cycle and successfully login with the new user. So it obviously worked fine.
In doing that power cycle, the board still did its 3.5 seconds of life thing.
But here's where it gets interesting: a couple of power cycles later, and the board stayed on! A tour of logs / sensors in IPMI suggested things were running (IPMI confirmed that the power state was "on"), but there were some abnormalities (quite a few of the sensors weren't running: some voltage, CPU / MB temperature). OK. Another power cycle, and the board stayed on again, and this time all the sensors are good.
I haven't attempted to boot into FreeNAS yet - as part of earlier fault diagnosis I reset CMOS, so it just boots into the BIOS screen (which I can navigate just fine). Later tonight, I might run a memory test and then attempt to boot FreeNAS (although I won't be risking any of my data drives).
I still consider the board well and truly toasted - there's no way I'd trust it for daily service anymore. But this latest observation certainly provides a few more clues as to what might be going on here. For me, it seems to supports the current theory.
For general interest, here is a grab of some sensor warnings in IPMI (these still seem to be appearing). Also a lone entry in the system log in BIOS - no idea what it means. Let me know if you'd like to see anything else.
Oh and ASRock support? William is awesome - he keeps emailing me and trying to help. Global support? Dial tone. Radio silence. Nada.