Major difficulty restoring Time Machine backups/accessing .sparsebundle on newer Mac Mini
Time Machine backups cannot be browsed in Finder. You must view them through Time Machine.
If you are using a computer that is different from the one where the backup was made, there are additional steps. The Time Machine - Troubleshooting page explains further.

Time Machine keeps the backups for each Mac separate, and normally only shows the ones for the Mac it's running on (selected in the Time Machine Preferences window), even if there are other backups of the same or a different Mac on the same or different disk/partition.
To see these &quot;other&quot; backups, you need the Browse Other Backup Disks or (Browse Other Time Machine Disks on Snow Leopard or Leopard) option.  It's available by Alt/Option-clicking the Time Machine icon in your Menubar, or by control-clicking (right-clicking) the Time Machine icon in your Dock.
You'll see a selection screen showing all the mounted disks/partitions that have Time Machine backups on them.  Select the one you want, click Use Selected Disk, and you'll be taken to the normal &quot;Star Wars&quot; display, where you should see all the backups on that disk/partition.
You may not be able to restore such items normally (to their original location);  if not, you'll need to restore them to an alternate location per Frequently Asked Question #16.