When I use the Timeshift GUI to create a snapshot, it takes a very long time to complete. If I click Cancel, my desktop environment crashes and I get kicked to a black screen or a login prompt.
When I use Timeshift CLI, it creates the snapshot in less than a second, but the command waits for a long time afterward. However, it can be safely interrupted with control-C.
Hmmm... so reading up, Timeshift doesn't automatically delete manual backups. Have you tried removing your old manual backups and trying again? This is just a hunch, but I feel as though Timeshift is looking through all of your old backups after you manually create a new one in order to apply appropriate tags. I think that's why it's telling you that the maximum weekly backups have been exceeded; because its checking every one of them.
Honestly, I'm just getting back into the Linux game after a couple years so I'm probably off, but it couldn't hurt to delete a majority of your old, unneeded backups and check.
I tried deleting the old backups, it still takes about a minute to complete the snapshot. I think it only limits weekly backups, and manual backups are unlimited. Weekly backups are completely fine, and manual is where the problem lies.