Mac – Big Sur – Time Machine structure

Time Machine at the moment of writing doesn't support APFS for the backup destination (see this), while it can backup FROM an APFS or HFS+ volume. That's because the AFPS filesystem doesn't support directory hardlinks, which Time Machine heavily relies on.

Probably Time Machine needs to be rewritten to take advantage of the features you mentioned, which is something that can happen in future versions of macOS.

It turns out that when you create a sparse bundle disk image that resides on an APFS volume that it is ALWAYS formatted as APFS even if you specify the the format as HFS+ like this:

I created a few of these (just to be sure) and each one failed as a Time machine Destination.

As @user128998 pointed out in his comment Time Machine relies on a feature of HFS+ to do it's job and will not work on one formatted as APFS.

I had assumed that disk utility was creating an HFS+ formatted sparsebundle because I had selected that option when creating the image, as shown in the screencap above.

In frustration I did a Get Info on the mounted sparseimage bundle and it showed as an APFS volume. Grr... So I deleted it and created another one and was very careful to NOT select APFS.

When I mounted that sparseimage it too showed as an APFS volume.

Opening Disk Utility I selected the mounted sparseimage volume and reformatted it as HFS+. The operation took longer than I expected (3 - 4 minutes) but it was successful. And this time when I ran the command

sudo tmutil setdestination /Volumes/MBAir

It worked!

So maybe I found a bug in Mojave (14.2) with disk images and/or sparsebundles. Or maybe it is expected behavior and I didn't know it, either way this can be marked as solved.