Next I tried my previous desktop. Of look, no power supply. Easy enough to rectify.
Oh listen, beep codes. Turned our to be memory issue (seating dust, bad slot... whatever it was it's not happening now).
Booted and grub said it was Ubuntu 14.10! Wow, what a throwback. It turned out it was actually 16.04, but that's still pretty old. Worked fine though.
The CPU was an AMD Phenom II X4 965, which was from 2009. It's 3.4 GHz, but, unsurprisingly, it didn't have IOMMU support and so I couldn't run Qubes on there. The installer didn't even manage get into grub. Some error about i386 modukes not found (which is peculiar, since this is a 64-bit CPU).
So the search continues...
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