EDIT: I mean it'd probably be OK if I were just shooting a video--I can fumble through it connecting unreliably, or just do another take if it unexpectedly disconnects. But I'm doing multiple video calls a day, and the webcam built-in to my monitor "just works" every time. It's tough to justify the additional complexity for something that isn't working reliably.
That's why I started leaning toward "Clean HDMI". With that method, as long as the HDMI capture device works, everything should "just work" on the Mac side, and as long as the camera can output clean HDMI, it should also "just work." I'm not dealing with a poorly-supported software webcam utility, special USB signaling, or annoying inactivity timeouts.
But it looks like I'll need a different camera, and it won't be cheap, but at least it's an option.