I predict that using your backup drive will go faster and be easier.
Do you have the new Mac yet?
Suggestions:
On the day the NEW Mac is scheduled to arrive, run "a last backup" on the OLD Mac.
When the new one is there, take it out of the box and set it up.
Connect your backup BEFORE you press the power on button for the first time.
Begin setup. When setup assistant asks if you wish to migrate from another Mac or drive, YES, you do.
"Point the way" to the backup drive and give SA time to "digest" things.
OK, here's something to consider:
You haven't told us HOW OLD the "old" Mini is, or what version of the OS is running.
If it's VERY old (say, 2014 or earlier), you probably want to leave most of your old applications behind, as they may not work on the new Mac. So... I would UNCHECK the option to "migrate" all the apps, and reinstall them later as needed.
If the version of the OS you're using is more "recent", you might migrate the apps along with everything else. But be aware that some still might not work with the new Mac.
Once you have this decided, I'd migrate everything else.
Just turn setup assistant loose and let it do the job.
When done, you should see your old login screen, just as before.
Good luck.