Time Machine forgets disk, does not execute, or executes mysteriously

On Ventura and for a long time, backups have become impossible to manage. I thought it was the drive and I purchased a new one and overkill. I formatted a number of times through Time Machine itself.

The geist is:

• Time Machine forgets the disk. If I 'add disk' and select my back up disk, then the disk shows in the list in the TM configurations. If I close the config window, when I open it again, the disk is not there. If I wait long enough with the config window still open, as I did while typing this post, the disk disappears from the list in the meantime.
• Time Machine does not execute. It does not back up, even if I select manually 'backup now', when the drive is present in the config window list, nor if I select 'backup now' from the context menu of the drive icon in the finder (including desktop).
• Time Machine backups are not accessible from TM itself. I can't enter TM backups from menu bar drop down menu, because it does not believe there is any backup disk to browse. However, if I open the backup drive through a finder folder, the finder offers me the option to browse my backups, and at least at a surface level, it seems to work. Finder sees the old backups and is able to make them accessible.

Note:

• All along the drive is available and is recognised as a backup disk type by the finder.
• Rebooting does not work.
• Disconnecting before rebooting, re-connecting after rebooting does not work, nor any other kind of dark art makumba ritual.
• My backup drives both work, file systems are fine, the problem comes back immediately if I format them and start from scratch.

My provisional conclusion:

• The Finder side of things works and finder does what it is supposed to do with the drive. Drives are fine. So it is solely the TM code/application/GUI side of things that does not work and it shows as a memory/execution problem.

Is there a way to hard reset the TM settings or any other 'deep' setting that I can do something to?