Myself & another Mac User Group member had been offering advice to a third member on how to get an external HDD used for Time Machine backups to be recognised by the Finder on his iMac. We went through various options and as yet have not heard back from the fellow.

Last night I attempted a Time Machine backup of my 2017 MacBook Pro and experienced exactly that problem! My MacBook wasn't recognising the external drive (USB-C connection) in the Finder. I went through all the recommendations we'd given the other fellow with the same problem. I even opened up the drive case to ensure the internal SATA connection was intact. It was fine.

Time for some diagnosis, I thought. Rebooted the Mac & started it in Recovery mode (⌘ & R keys at startup). Immediately the connection LEDs on the external drive illuminated, showing the drive was now making connection. No longer needing the diagnosis from the Recovery drive, I booted the Mac normally and found the external drive was being recognised.

I suspect that had I rebooted into Safe Mode a similar result would have ensued. Sufficient to say the drive behaved normally & another Time Machine backup succeeded.