If a clean install of macOS does not solve the problem, then I suggest "Restoring" the firmware procedure as outlined in this Apple article:
Revive or restore an Intel-based Mac using Apple Configurator - Apple Support
Once the system is updated, the macOS installer will not update it again. This can be seen when people use a third party internal SSD with the Monterey installer. Monterey will not install if the internal drive is a third party SSD. Once Monterey is installed with an internal Apple OEM SSD, then it is possible to install Monterey with a third party internal SSD installed. The macOS Monterey installer will not install with a third party internal SSD if a system firmware update needs to be applied.
The "Restoring" the T2 firmware procedure is for resolving "odd" situations which affect booting & installing macOS. It is basically the most drastic means of solving any 2018+ Mac issues. If a "Restore" procedure does not solve your problem (you must test the Mac before installing any third party apps or migrating/restoring from a backup, and with no unnecessary external devices connected), then it indicates some sort of hardware issue. Restoring the T2 firmware erases the internal SSD and pushes a clean OS onto it.
FYI, When M1 Macs were released there were reports that some Samsung T5 SSDs were not compatible due to the USB chipsets the drives used (T5 SSDs utilized more than one version of the chipset). Plus macOS Monterey is known for introducing compatibility issues with various minor point release updates. Make sure to perform a clean install of macOS to the external drive by erasing the whole physical drive as GUID partition and APFS (top option). Within Disk Utility you may need to click "View" and select "Show All Devices" so that the physical drive appears on the left pane of Disk Utility.
You may also want to check the health of the external SSD. Maybe it is some sort of problem. You can check the health of the SSD by running DriveDx and posting the complete text report here using the "Additional Text" icon which looks like a piece of paper.
Have you tried using any other external drive?
Does this T5 SSD have USB-A/B ports or USB-C ports? If you are using any adapters to connect to the Mac Mini, then maybe the adapter is the source of the problem.
FYI, Macs have always been picky about the external drives used for booting a Mac. Years ago I encountered a USB drive enclosure that would not work at all with a Mac, but worked perfectly fine on a standard PC.