If the internal hard drive has failed, it could possibly be interfering even when booting to external media.
Personally I would create & use a bootable Knoppix Linux USB stick to attempt booting that old Mini. Knoppix is better at ignoring some drive failures since it doesn't try to mount any volumes unlike macOS which will try & try and even get stuck in the boot process because macOS cannot handle the errors well. You can use the downloaded Knoppix .iso file as a source for Etcher (Mac/Windows/Linux) to create a bootable Knoppix USB stick. Make sure to download an .iso with "EN" in the name for ENglish. Option Boot the USB stick and select the orange icon labeled "EFI".
Make sure to give Knoppix lots of time to boot since the Mac may appear frozen while booting Knoppix until it reaches later in the boot process (issue with Mac firmware not being friendly). If Knoppix boots to the Desktop, then click the "Start" menu on the lower left of the Taskbar and navigate to "System Tools --> GSmartControl". Post the complete GSmartControl report for the internal drive so I can review it. It has been a long time since I provided these instructions so I forget exactly how you access the full health report for the drive (I know there is a button for a "view" option and one to "save" which will provide you with the full report.
