Thankfully I have another unused desktop with hardware from the same time period and I was able to take the power supply from it, pretty much the same spec. My desktop is fortunately back up and running now. Still, I'll have to buy a new power supply if I ever find a use for that other desktop. Unless it's just a blown fuse or something, but I don't know if I want to attempt repairing a high-voltage device.
Be careful with the mains voltage part of a PSU! I've replaced two big 400V caps in the internal power supply of a DVD player. It worked for an hour. When I turned it on again the whole thing went boom. What did I do wrong with the solder work? Probably nothing. Two of the four diodes in the rectifier decided to identify as simple wires coincidentally.