(and you know that it is rare that the PSU dying doesn't kill the MOBO or something else).
Actually, it is rare, but not unheard of. Typically, when a PSU dies, it does
not do so catastrophically but rather, one or more the output voltages stops - stopping the computer.
Sadly, what typically happened when more than just the the PSU fails is there was a severe power anomaly that takes out the motherboard (or something mounted on it) and the PSU at the same time.
That said, a cheap, no-name, budget PSU will not have advanced safety and protection circuits as a maker's higher-end models will either. If using a cheap PSU, all bets are off.
You can test RAM using one of the following programs. Both require you to create and boot to a bootable floppy disk or CD to run the diagnostics. Allow the diagnostics to run for several passes or even overnight. You should have
no reported errors.
Note, however, that software based RAM diagnostic tools are good, but not conclusive. So you might try running with just a single RAM module to see if it fails. Repeat process with remaining modules, hopefully identifying the bad stick through a process of elimination.
It appears the download link to Microsoft's tool is broken. I have reported it and hopefully it will be fixed soon. But the Memtest link works, and the native (built in) tools don't require any download.