What you are describing is not the normal Monterey installer process.
On a Late 2014 Mac mini, booting from a valid Monterey installer USB drive should normally allow you to select the installer by holding the Option (⌥) key at startup.
After selecting the external installer, the Mac should boot into the recovery environment contained on the installer itself and then display the macOS Utilities window, where you can choose Disk Utility to erase the internal SSD or Install macOS to begin a clean installation.
The fact that you are seeing Recovery Assistant instead of the macOS Utilities window suggests that the Mac is not actually booting from the installer, or that something is preventing the installer from loading correctly.
One possibility is that the Mac is booting from its internal Recovery partition or Internet Recovery rather than from the USB installer.
If the USB installer is not recognised as bootable, the Mac may fall back to another recovery environment.
Another possibility is that the USB installer was not created correctly.
If it was made using a third-party utility rather than Apple's createinstallmedia command, it is worth recreating the installer from a fresh copy of the Monterey installer using Apple's recommended method.
It is also worth resetting the NVRAM. Shut the Mac down completely, turn it on while holding Option, Command, P and R together, keep the keys held for about twenty seconds, and then try booting from the USB installer again by holding the Option key during startup.
The behaviour you describe, in which the Mac eventually appears to enter sleep and cannot be awakened, is not normal.
This could indicate a display issue, a problem with the USB installer, or a hardware fault. If possible, try using a different USB flash drive, a different USB port, a different monitor or display cable, and a wired keyboard.
Sometimes the computer continues running while only the display fails to wake.
A failing internal SSD can also interfere with the recovery process.
If you are able to open Disk Utility, check whether the SSD appears normally.
If the drive is missing or Disk Utility becomes unresponsive while accessing it, the SSD itself may be contributing to the problem.
The Late 2014 Mac mini is officially supported by macOS Monterey, so compatibility should not be the issue.