it is clear that without Linksys or Apple help, the days of Time Machine for me are numbered.
Not sure what kind of "help" that you are expecting from Apple. They are not going to be able to help with an issue on a Linksys product.
If Linksys advertises "Time Machine support" for their products, I have not been able to find it anywhere. Either Linksys supports Time Machine backups to a drive at the USB port on their router, or they don't. If they do, it's got to be in print somewhere.
What did Linksys Support say in this regard when you contacted them? If the finding turns out to be that Linksys does not really officially support Time Machine at the USB port on their routers......as I suspect.....you have your answer.
CCC does have some limited ability to store and create a history of deleted files on your Mac. Check out Safety Net.
If you really do need Time Machine, backups to a hard drive at the USB port of the Synology router have been flawless for me for well over 2 years. Asus and TP-Link also advertise Time Machine support prominently in their specifications.