So. After the remedial work below, I fully expected the problem to be fixed. But no! On logging in the next day I see that the QNAP lost power at 3:09 in the morning. It reset, re-synchronised the RAID array and reported the filesystem corrupt. There was certainly no-one unplugging it at that time of the morning unless we have a secret midnight cleaner.
A quick piece of online searching reveals this to be a common problem with some QNAPs and some versions of firmware. Mine is an SS-439 on 4.2.6. I don’t care what models and firmware they say, it looks like the PSU just gives up under certain conditions. This is bad because it means faulty hardware and it’s hard to do anything about that. But, more research needed.
P.S. I switched the PSU for the one from my old QNAP a couple of days after the above.