Never mind, I guess(?)
I tried with my desktop PC's monitor (A 32" Samsung). Took a bit of work since it's mounted in a rather nonaccessible fashion (and uses Displayport for my PC so I didn't have a cable end hanging). Anyway, I attached the cable/adapter to it and....HDMI did not work.
Tried again, secured all connectors.
Turned off Wifi just in case casting would interfere with HDMI out.
Finally, flipped the USB-C connector to other polarity. Lo and behold, immediately started working.
1920x1080, 60 Hz, so essentially nothing special.
Flipped the USB-C connector *again* and...it stayed working. No matter how many times I unplugged the adapter and put it back in. Turned of Wifi. It still worked.
Afterwards, I tried all the same combinations with the LG TV again. No-go. So there's I guess *some* compatibility issue with, well, some displays.
So...essentially, I've confirmed that the adapter does work, at least with a single display, but I really wish I knew what are the requirements for the HDMI to actually work. Is there some bad soldering on the connector, or what exactly is going on?