I read at [1] (see at bottom) that there was a problem with xhci in kernel 5.10 (which was fixed in a later kernel than 5.10) which could cause these USB cards to not work. So, I compiled the latest vanilla kernel (5.16.11) from kernel.org and booted up Bullseye with it.
The boot did not succeed 100% (some other things broke, such as lightdm), but it did bring up the networking correctly and gave me access to a shell. I was able to verify that the ethernet port on the USB card worked fine. Also, the "xhci_hcd 0000:1c:00.0: WARNING: Host System Error" log was not present and neither were the ax88179_178a kernel failure logs. So, it looks like whatever kernel software changes fixed the xhci also address the ethernet port issue (or perhaps the entire card in general because now I realize even the USB ports on it were not working in kernel 5.10). However, I have no idea what the kernel software changes that fixed this are, and I have less of a clue as to how to figure that out. [1] = https://community.ipfire.org/t/update-158-to-161-problems-with-usb-ethernet-adpater/6854/5
