Following on from the current end-state of "JACK starts if the interface is set to hw:PCH or hw:PCH,0, but I don't have any speakers except on my screen, which is on my graphics card's HDMI, so I'm not sure there'd be any point.", I:
- Found some terrible old headphones
- plugged them into my computer's built-in headphone jack
- ran
a2jmidid -j default - set JACK's interface to
hw:PCH - Hooked up various connections in JACK following this guide
- successfully recorded some random noises on the (MIDI) keyboard
- played them back through the headphones.
So my remaining issue is that it's still producing the error in the question when I try to point JACK at my graphics card's HDMI audio-out instead. Since I don't have separate speakers or other not-built-into-my-monitor audio output, that's still a problem, for which I will leave this question open. In the event that nobody has an answer for that, hopefully this answer will at least help other people without my restrictions get set up.