My Disc Utility does allow the erase function on my mac mini. How can I proceed?

Which drive are you trying to erase? An external drive or an internal drive? You cannot erase the same drive you have booted from. You need to boot from some sort of external media such as a bootable external macOS drive, or a bootable macOS USB installer, or Internet Recovery Mode.

If you are trying to erase an external drive or are having trouble doing this on the internal drive when booted from external media, then make sure to click on "View" within Disk Utility and select "Show All Devices" so that the physical drive appears on the left pane of Disk Utility since recent versions of Disk Utility from macOS 10.13+ now hide the physical drives from view by default.

If you are running macOS 10.6 to 10.10, then the method of erasing the whole physical drive is a bit different as it may require using the "Partition" tab within Disk Utility.