No Time Machine backups b/c "sparsebundle not accessible"

Ahhh, thank you very much! That is truly helpful.

So I had a pretty good gut feeling, because I bought a new additional SSD external drive at the end of last year and connected it directly to the iMac's USB port, "just in case". This one works fine and I disconnect it after each daily backup so it can not be compromised by any "outside" intrusion. Something tells me that the possibility for a tech virus for our devices like Corona was for us humans is pretty high and not too far fetched, so I disconnect my backup devices now regularly.

And your answer even does help a lot, because this WD Drive failed several times over the last years and I was always able to make it work again, but I also knew it would/might completely fail one day. Even WD stopped updating this old device last year. But heck, I thought, well, it is still working, it has 4 TB of storage, so I will use it "just" as a backup device. So it was mumbling away every day, making sounds like "doing something" but I realized it stopped working two days ago and the iMac gave me this error response.

So thank you again for somehow confirming a lingering problem to me.

Hahaha, oh yeah, you are so right 😂🙈 but I had so much hassle with EHD‘s … I have at least 4 of them who quit working after a very (!) short period of time. It was devastating. Only the WD MyCloud was working really well over the years - and I tried a lot of others before, like WD Passport for Mac etc. They all failed way too early. So now I chose SSD and hope it will work a bit longer 😅👍🏼