Time Machine Backups failing in Catalina (Time Machine must create a new backup for you)

I have exactly the same issue. But, I have used Synology successfully for years as a a NAS for Time Machine backup from multiple Apple devices. I only upgraded two of them to the latest Catalina release and presto, it does not work any more.

Apple has clearly broken something as my non upgraded Catalina systems are still working happily.

I think you're missing the point. Complaints regarding unsupported NAS failures to back up or restore typically arise with every macOS upgrade and even some incremental updates.

In all the years I have been using Apple products I have never known them to release substantive details regarding updates to Time Machine. The most I have ever seen are vague comments regarding "increased stability" which have no practical value. Then, bear in mind Apple tests and evaluates their proprietary software on Apple equipment. They have no knowledge of non-Apple NAS device operating systems or their file storage methods and would no interest in them even if they did.

Time Machine is exclusively Apple's IP. They disclose little to no information regarding it, and whatever they did publicly disclose in the past has not been updated in many years. Every non-Apple NAS vendor claiming Time Machine compatibility requires the motivation to release software / firmware updates to their products concomitantly with Apple's macOS updates. They don't. If anything, they're always months or even years behind. Why should they care? They got your money; their job is done, and if anything goes wrong they conveniently blame Apple. Or you, or your network, or anything else—anyone but themselves for irresponsibly claiming Time Machine compatibility.

Meanwhile, you're comfortably thinking your Macs are being securely backed up, when they're not.

I agree it shouldn't, but Apple's TC was designed from the outset to use AFP which has been deprecated. Using it as a client on a non-Apple network complicates things since AFP is a proprietary protocol that Apple never licensed to anyone.

It should work but all my networks are exclusively Apple which is the reason I suggest trying it. My experience attempting to use a non-Apple NAS with Time Machine was identical to yours. I gave up. After conclusively providing to Synology it was not reliable and could not be made to be reliable they refunded my money.

You can also use an external USB-connected hard disk drive connected the the TC. Or as many hard disks as you want, provided you use a powered hub.

I even binned my lovely Time Capsule for a new Synology NAS

That's too bad. A Time Capsule will work, whereas a non-Apple NAS will not. Never has, never will.

Please search this site for hundreds or thousands of identical complaints. Start here: Time Machine Keeps asking me to create a new back up.

As I understand it, your TC is a (wired) client on your network. Is that correct?

That should work, provided it's configured properly, but try using the TC as your router instead of a client on your existing network. Use it in that capacity, and then determine if TM backups continue to work. You probably ought to start by reconfiguring the TC and erasing it, which are two separate steps (resetting the TC to factory defaults and reconfiguring it does not erase its backups).

If necessary, the TC's hard disk drive can be replaced, with difficulty.

I have merely proven my initial post/point, that a recent update had impacted the ability for TM to successfully complete.

macOS upgrades and updates do not cause Time Machine failures, but they will reveal inadequacies in unsupported configurations. The additional hard disk drive activity associated with large file system changes can also cause a failure of a disk that was about to fail anyway.

But if you're trying to fix a failed Time Capsule by replacing it with a non-Apple NAS, that won't work.