It’s still visible in ALSA.
I think PulseAudio selects devices to show based on their names… and the device just didn’t get a proper name.
The fix is probably to use a USB headset, because that introduces a new sound card into the system, and it probably has the right name.
Of course, this isn’t necessarily going to work. The linked debugging page introduced a tool called codecgraph that creates a visualization of your computer’s audio inputs and outputs. Here’s mine:
Right click and look at it in full size.