"Is there a way to restore the Time Machine on a new Mac Mini to an external drive?"
I think this could work, BUT ONLY if you install a copy of the OS onto the external drive FIRST.
Not really hard to do:
1. Get the external drive
2. Boot to INTERNET recovery -- Command-OPTION-R at boot.
3. Format the drive using disk utility (High Sierra and earlier, format to HFS+, Mojave and later, format to APFS)
4. Run the OS installer, and install to the EXTERNAL drive.
5. When done, connect the tm backup
6. Start clicking through setup, when setup assistant asks if you wish to migrate from another drive, YES, do that
7. Aim setup assistant at the tm drive, and give it time to "digest" everything.
8. I would suggest you migrate everything, and let setup assistant do its thing (it will take a while).
9. When done, you will have a bootable external drive, with everything on it.
At this point, there are still steps you will have to take to do an "additional migration" to get SOME of the data to the new Mini, but you need to get the data OUT OF the tm backup and into "finder format" first.
SOMETHING ELSE you may need to do:
DO THIS BEFORE YOU DO THE ABOVE:
Boot to recovery. I believe with the m1 Mini you need to hold down the power button for about 4 seconds to get to the startup options panel.
Then, "work your way" to the recovery partition.
You need to get into "startup security", and DISABLE all protection there, so that you can boot from an external drive if necessary.
I disable startup security as a matter of course. One doesn't need it.
One final thought:
If you had been using either CarbonCopyCloner or SuperDuper as your backup utility (to create cloned backups), things would go much easier!
