It’s been clear, the issues are on the boot disk. “Simple” solution (IMO):
1. Buy the external SSD (I used Samsung T5) for bootable disk. Attach it to the back of iMac screen (I used 3m double tape)
2. Install the Os on the external SSD
3. Check if the problem remains (very unlikely)
4. Reformat the original disk, check whether any parts of it are still usable (whether you can store the data safely)
5. You can restore or replace the original disk (I don’t know whether this is possible for your iMac). The replacement disk shouldn’t have to be your bootable disk (since you have one anyway). It also can be non-SSD (cheaper) because all the swap and temporary files are already in the new external SSD.
6. Other option, the new external SSD (in step 1) - get bigger capacity (around 3Tb to match with the original disk)
Add: answering your question. It’s not worth to try with other (newer) OS. Failed disk will be worsen on the newer OS (due to they are more complex - need more ram, more swap, more cpu etc).
Monterey isn't newer. Until Monday, I was running Ventura with no problems.
I'll have to put the drive inside or remove one of my external monitors which I don't want to do.
Thanks for all the help everybody!