OP:
Time to seek out "live" help.
Either at a brick n mortar Apple Store, or perhaps a 3rd-party independent service provider who works on Macs.
If you have to "carry it in".... then... carry it in.
I have never ever (in 35 years) seen a Mac that couldn't connect and copy files to an external drive. Even when it wasn't working right.
Not ever.
Hi. Yup, macOS Monterey's (I remember Big Sur having a problem with mounting too but rarely) the very buggy culprit, at least for older external drives and relatively older Macs. I'm tempted to just fresh install everything, start everything from scratch (all those decades of work- but letting go is peace sometimes
, forget the backup, a lot of the documents and files they're irreplaceabe but not life or death, then just go back to High Sierra (everything will work there for sure, like it's working now on Windows Bootcamp or on my PC backup), start anew, as if it was the first time using a Mac or computer again.
I did buy four 2TB SSDs (MX500) just today, brand new for just US$18 each free shipping in that strange company called 'want.jp'(theirs names may be different in every platform in every country but it always has the 'want' and 'jp' coz' it's a japanese company with a unique business model, the items come from Shibuya, Japan- downside is takes weeks to ship, 3 weeks some say, at most) it's in every major online shopping platform covering all industries, including fashion and computers) that seems to threat the online shops like a casino or lottery to boost itself up (and it works, it's a top store worldwide).
The price is now back to normal, though at US$207 to $244- that was a huge blessing, divine providence level (it was by accident last night, I tried one last time for the lowest price as I was about to buy at this other shop). It was surreal, though it'll be considered clear once those four 2TB (total of 8TB SSDs SATA) are tested on Windows PC w/ SSD testing apps coz' it's extremely too good to be true. I just hope it's genuine and is really brand new, so it has the Crucial 5 year warranty (it can be refunded within 30 days but I'm deeply praying & hoping, that hassle will be avoided and it's really flawless, brand new and legit).
I'll open the iMac (I have the kit to stick the 5K screen back & an 1TB NVME bought last year w/ SSD stick adapter- there's that slight fear even today after a year of opening it up but courage is best, it's not like the old screen with magnets) try to back it up internal to internal but fresh install w/ High Sierra (Time Machine in Extended Journaled), once all files are in the 2TB, I used the other 2TB SSD external and back that up, then fresh install again High Sierra on the faster NVME and the 2TB internal SATA becomes the Windows Bootcamp disk (just games installed to be played on sundays, kind of rest after sunday worship/mass after all these complications)- it's a lot but w/ the extreme boost in bandwidth from a hard drive, it won't be so bad.
If Time Machine created by Monterey can't be restore to an older OS like High Sierra, I'll just copy paste the files manually (have done that before when Time Machine failed to restore- must be the layers of files of updates over more than 1 decade since Time Machine was launched) and reinstall the 3rd party apps that's compaitible w/ High Sierra at least.
Thanks for the reply and for the time, it's meditative to write when things aren't going well for a long time, if it's alright to share. Stay safe.