I checked with my neighbor........who did replace an AirPort Extreme with an attached hard drive with a Synology router and he has not had any issues with Time Machine backups. He added a new hard drive when he upgraded his Mac to Big Sur to start backups that way.
Because he did not even know that Big Sur changed the drive format for Macs, he went ahead and formatted a new hard drive the same way that he has done in the past using Mac OS Extended (Journaled).
Time Machine took care of everything else automatically. He just discovered by examining the sparsebundle file that holds all the backups that the sparsebundle was formatted in APFS (Case Sensitive), although the hard drive itself remains formatted in Mac OS Extended (Journaled).
So, his backups are formatted differently than the hard drive, a fact that he did not know. Frankly, this makes things more complicated and with that come more chances of errors.
If it works for you, that's great. If I were trying the same thing, I would also connect a hard drive directly to my Mac(s) every day or so and back up using Time Machine that way as well. It's a simpler and much faster process when Time Machine backs up to a "local" drive.
Note.....my neighbor only uses the hard drive that is connected to his Synology router for Time Machine. He does not use the drive for any other type of file storage, sharing, etc.
I'm not trying to defend Synology's support group, but to be fair, nothing has changed for Time Machine. You still format the hard drive the same way that you did before Big Sur and then let Time Machine take care of everything else.
Your mistake......and mine......was thinking that since Apple changed the format of the hard drive on Big Sur Macs, that a backup drive needed to be formatted the same way.
As a test, I am going to format a spare hard drive in APFS (Case Sensitive) and connect it directly to my Mac to see if Time Machine will back up to the drive that way. Will report on the results later today.