A mystifying fault, as you said! You could try plugging one end of the USB cable into a power bank and the other into the computer. If this does the same thing then the motherboard is simply reacting inappropriately to a USB connection whether due to a fault or a design decision. If the power bank does not cause a restart then my only explanation is magic.
I have now tried various other USB devices: my phone, a powerbank, my in-ear headphone case and an old external harddrive, which is the only other USB device I own apart from the UPS that has a USB B plug. None of them trigger the strange auto-reboot behavior. It is only caused by the UPS.
On a whim I fiddled around with more USB settings in the BIOS and turned off “USB Legacy Support”. This had the effect of my USB keyboard no longer working before TrueNAS is booted (so I can’t even get into the BIOS anymore until I dig up a PS/2 keyboard from somewhere), but the UPS still triggers the auto-reboot.
I also managed to track down Eaton’s (Windows-only 
) UPS software (SetUPS as well as UPS Companion). UPS Companion can see the UPS but does not allow any kind of configuration of the UPS itself. SetUPS also makes something happen (I can hear the UPS click its relays when the software starts up), but the software outputs some random error message pertaining to the device no longer being supported. I was unable to track down an older version of the software.
At this point I might just have to bite the bullet and buy a new UPS. Any recommendations for like a cheap UPS with USB control? I am reluctant to just buy the followup Eaton product, fearing it might just have the same issue.