I downloaded the vanilla 5.16.11 kernel from
kernel.org, compiled it and installed it. This time the resulting kernel worked flawlessly (well, almost). The machine boots normally and the USB card is working great. Both the ethernet ports and USB ports in it are functioning well. The only issue I have found so far is that I am not able to mount ISO9660 partitions, but besides that everything is perfect. This filesystem issue is probably an unrelated issue...
Anyway, I decided to check the Debian backports repository and noticed that the linux-image-amd64 is using debian kernel 5.16.0, which I think maps to vanilla kernel 5.16.12. So I proceeded to install that backports package. I did that because I would like to run a debian-supported kernel rather than a self-compiled kernel.
Unfortunately, the USB Card problem came back when I installed the debian backports kernel. The xhci_hcd log stating that a "Host System Error" occurred showed up again in /var/log/messages, the ethernet port on the USB card no longer works and also the USB ports on the card do not work either.
So, my conclusion is that there is something specific to the debian kernels that is breaking this USB ethernet card.
For now, I am going to stick to the vanilla kernel I compiled myself, but it would be great if the debian maintainers did something about addressing this issue at some point (hopefully sooner rather than later!) so folks like me can go back to using debian-supported kernels.
So far I have not seen any maintainer chime-in here though. I do not know if it is because I am not providing the correct information, if they have bigger fish to fry or something else... I still offer my help though to try things out to assist the maintainers at getting more information (or the right information!) or to test patches or whatever...