How to prevent Time Machine from formatting drive to APFS. MacOS Monterey.

ERASE the drive, again, and deliberately choose the desired format needed, rather than taking the defaults. exFAT probably provides support for the largest files without getting into proprietary formats like NTFS.

Once connected to your Router (PROVIDED it supports Mac backups) your Mac will create a sparse bundle disk image (a Volume inside a file) on the remote device, and a Mac file system is created inside, holding your backups in the the sparse bundle-disk image.

To the remote device, the sparse bundle disk image is just a file, that occasionally grows in size.

To the Mac, that is a remote shared backup destination.

'Remote' in this case means not directly locally attached (i.e., not a local USB drive any more).

<< [time machine]...Wouldn’t even see the existing backup. >>

remote backups are not using the same format as local backups, so a backup made locally cannot be re-used as a remote backup.

if you made certain to Mount the pre-existing remote backup drive so that it could easily be found, Time machine should have offered the option to "inherit" the old backup.

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Going forward, you are not going to use this in the expected, ordinary way. That will require you to manually format (actually bundled into ERASE command) the drive first, in a format that the Router can use, then connect it to the Router and verify that the Router sees the formatted drive, then tell time machine to place its backups there (when it is ALREADY "remotely" connected.

When you tell time machine to use a local drive, it will re-format in the format it prefers, reducing your options.

This may help:

Types of disks you can use with Time Machine on Mac - Apple Support

Regards.

redacted.

...was same info as users @varjak paw and @ Grant Bennet-Alder.