Anyway, it was brought to me displaying out of ink, but the cartridge was caught on something and couldn't be removed from the printer. I did a bit of searching and found some forum posts with the same issue, but the official answer was, as expected:
Which is, of course, a decent first step in many situations, but also doesn't usually fix mechanical issues. One user vaguely mentioned a loose gear somewhere within the machine, which was more promising than other comments of "I used pliers to force the cartridge out and it broke, and now my printer still doesn't work" and at least gave me something to look for, so it was time to dig in.
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(Minor note: I didn't think to take photos until I had already found the issue, so we're pretending the reassembly photos are disassembly photos).
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On the bottom, there are 4 tiny screws that hold the case together, then 3 larger ones that hold the mechanism in the bottom of the case.
Taking them out and carefully lifting the top of the shell, I found and disconnected the ribbon cables for the screen & control panel.
Then I disconnected the fan (red & white wires) and battery connector (another ribbon cable) to remove the back panel, which was just slotted in and slightly trapped under the PCB.
The mechanism still won't come out of the case because of the PCB (the USB ports were catching) so I took out 3 tiny screws and was able to shift it a little to angle it all out.
If I knew what I know now, I could have skipped taking the PCB off the mechanism, but since I didn't yet know exactly what I was looking for, there were more ribbon cables and connectors to deal with.
After moving gears, turning it over & over and poking at various things, I finally found that the cartridge was trapped by a pressure bar, which lowers to press the ink film against the paper, but failed to retract. Unfortunately this was before I thought to take pictures so I don't have a shot of the mechanism, but this is how it's supposed to move:
I manually rotated the gears and released the cartridge (which I also didn't take a picture of) then traced the mechanism till I found the culprit: the drive gear had slipped to the end of the motor shaft and no longer had enough friction to be turned by the motor. So I put a tiny dab of super glue on the shaft and pushed the gear back into position, then put it all back together.
I powered it off and on a few times to be sure the pressure bar mechanism worked as it should (same video as above), then put the cartridge back in and was able to print (back to the first picture again).
All in all, it took about an hour, including downloading and reading the user manual to get an idea how it was supposed to work. It's a pretty neat little printer and surprisingly easy to take apart.