Mac doesn't recognize external hard drives after being installed

Make sure all checkboxes are checked in Finder's General settings pane:

Using Disk Utility, select the external drive that will not mount > click First Aid > click Run and report what happens.

see > Repair a storage device in Disk Utility on Mac - Apple Support

How are the external drives formatted?

Try disconnecting everything and resetting the SMC.

see > Reset the SMC of your Mac - Apple Support

Desktop computer

1. Shut down your Mac, then unplug the power cord.
2. Wait 15 seconds, then plug the power cord back in.
3. Wait 5 seconds, then press the power button to turn on your Mac.

Then reconnect the external drives one at a time.

If that doesn't help, try resetting the NVRAM.

see > Reset NVRAM on your Mac - Apple Support

How to reset NVRAM

1. Shut down your Mac.
2. Turn on your Mac, then immediately press and hold these four keys together: Option, Command, P, R.
3. Release the keys after about 20 seconds, during which your Mac might appear to restart. For example, you might hear a startup sound more than once, or see the Apple logo more than once.
4. When your Mac finishes starting up, you might need to adjust any system settings that were reset.

Disk Utility should show how the external drives are formatted.

For example the Kingston DataTralever 2.0 that I had laying around is formatted Master Book Record / MS-DOS (FAT32)

The other external drives that I use for backup and files are formatted GUID Partition Mac / APFS or Mac OS Extended (Journaled)

Then the external drive is toast.

The current version of Monterey is 12.6.2

see > Find out which macOS your Mac is using - Apple Support

and then to update your iMac, see > Update macOS on Mac - Apple Support

How are they formatted?

You’re welcome.

Sorry to hear that you couldn’t repair it.

FWIW I always keep multiple backups of critical data.