Time Machine is NOT a sparsebundle

I suspect that the drives are formatted in APFS format:

Currently, when backing up to HFS+, Time Machine creates a folder within which is a replica of the file system of the volume being backed up. Files and folders which haven’t changed since the last backup are represented there by hard links: inspect the inode number of two corresponding ‘copies’ of a folder which hasn’t changed between backups, and you’ll see that they are one and the same, with identical inodes, just as regular file hard links are.

APFS doesn’t support directory hard links, so can’t use the same mechanism when storing Time Machine backups. Instead, what appears to function as a form of virtual file system is created using new features in APFS. The volume assigned the role of Backup appears to be a regular APFS volume, and is protected from normal access, even by root. File data is kept as usual in the container’s Physical Store, to which file data is copied during each backup. Apple hasn’t indicated whether this continues to be whole files, or whether only changed data are copied.

This is synthesised into what is presented by the Finder as the customary hierarchy of files and folders, just as with HFS+ backups. However, matching unchanged folders have different volume numbers, as if they were stored on separate mounted volumes. Their own inode numbers and attributes are identical, though.

Above from:

https://eclecticlight.co/2020/06/29/apfs-changes-in-big-sur-how-time-machine-backs-up-to-apfs-and-more/

I do not know what to do. Time Machine can be hard to get to work on drives that use their own apps like NASs and Drobos.

I would use whatever works. I wouod use the latest Apple wants to use to help avoid problems in the future

The backup scheme in Monterey for Time Machine is significantly different than in the past.

Time Machine now requires an APFS formatted drive as it leverages a feature of APFS called snapshots. Instead of entire files being backed up, a snapshot of only things that have changed is made. So, when restoring a file from a backup, Time Machine uses the snapshot feature to recreate the file. This process is much more space efficient.

So, the backups stored by Time Machine are not a traditional "filesystem" as it once was. As a matter of fact, hourly snapshots are made of the system drive on the drive itself as well when using Time Machine. Only a 24 hour period is maintained so as not to fill up the system drive. The other case, any macOS update will create a full snapshot of the system which allows faster restores. This too will disappear after twenty four hours releasing the space.

You did a direct backup to an external drive. Sparsebundle disk images only occur with a network backup.

In Big Sur and later, a direct backup will be formatted APFS (Case-sensitive), and you will not have any other access to the drive.

You would need to partition the drive to handle multiple direct backups or other usage. It may be possible to use Volumes instead of a partition, but when I tried on Big Sur, it didn’t work as I thought.