PSU is fine
OK, so I had to learn details about ATX power supplies the hard way, didn’t I. ATX PSUs do not power up if there is something fishy with the motherboard. In this case, it included an extra USB connector from the case. Turns out that it’s not enough that pin diagrams match, they also have to match reality.
My PC now works fine, it’s considerably quieter and cooler than before. Restoring stuff to a state resembling the way it was before the hard drive crash will take weeks. I will have to get some dedicated hardware for backups in the future, the ones I had were broken or incomplete. The Buffalo Terastation looks like an interesting candidate. 750 GB of RAID5 on a gigabit NAS unit.