Following this thread with great interest. I'm not a tech guy; I'm just a long time Mac user. I'm not certain this problem is confined to Monterey.
I've got a 2018-ish MacBook Air, which is still running Mojave. Last month, one of my external hard drives wouldn't mount on it; it showed up (greyed out) in Disk Utility, but I couldn't run Disk First Aid on it. Tried everything - new cable, restarting, different ports, even a different (and older) MacBook. Two days later, nothing could make it mount. Thought I'd have to go to a data recovery place, but figured I'd try Disk Drill first, which had been recommended to me. (It's not supposed to fix a damaged drive; it's supposed to help recover data.) With the free version, I could see that all the files were still there, so I bought the full version, and started the recovery. Out of the blue, the drive appeared again, mounted on the Air. Drive has been fine ever since.
Cut to last week. I bought a new Mac Mini (M1) which, of course, runs Monterey 12.5.1.
Same thing happens with a brand new external drive from Seagate. Wouldn't mount on the new Mini, or the Air running Mojave.
Ran it through Disk Drill, and it mounted after several minutes.
No idea if using Disk Drill actually did the trick, or if it was just a coincidence.
Gotta say, this problem makes me really skittish.