1) Automatic Computer Sleep Mode is causing a sleep/shut down problem for some setups.
As a test:
Turn On "Prevent automatic sleeping when display is off"

Set "Turn display off when inactive" to 5 or 10 minutes

Leave the Mac Mini On and only allow the display(s) to turn Off
Do not manually Sleep from the Apple Drop Down menu
Shut Down the Mac when away for a day or more
I wouldn't count on it, but solution may come if enough folks send Feedback.
Feedback - Mac mini - Apple
Feedback - macOS - Apple
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2) If your Mac mini is running 15.5 as shown in your profile..?
Updating to the latest macOS 15.7 could possible fix the sleep/shut down issue.
What's new in the updates for macOS Sequoia - Apple Support