What does softwareupdate via the Terminal output in you Mac mini?
Maybe you could make a Big Sur bootable installer and try to Option-boot it -- if it boots then it might well install it (my Mac mini boots such Mojave installer but fails to boot High Sierra installer).
My Mac mini 2018 has the following output:
softwareupdate --list-full-installers
Finding available software
Software Update found the following full installers:
* Title: macOS Monterey, Version: 12.1, Size: 12157035487K
* Title: macOS Monterey, Version: 12.0.1, Size: 12128428704K
* Title: macOS Big Sur, Version: 11.6.2, Size: 12433351292K
* Title: macOS Big Sur, Version: 11.6.1, Size: 12428472512K
* Title: macOS Big Sur, Version: 11.6, Size: 12428553042K
* Title: macOS Big Sur, Version: 11.5.2, Size: 12440916552K
* Title: macOS Catalina, Version: 10.15.7, Size: 8248985973K
* Title: macOS Catalina, Version: 10.15.7, Size: 8248854894K
* Title: macOS Catalina, Version: 10.15.6, Size: 8248781171K
* Title: macOS Mojave, Version: 10.14.6, Size: 6038419486K
Then I can download the latest Big Sur:
softwareupdate --fetch-full-installer --full-installer-version 11.6.2
And make a bootable installer from that:
https://support.apple.com/en-us/HT201372