I saw this post in the linked thread: Plasma 6: Task manager doesn't save new apps - #10 by bluebomber182
I can confirm this worked. Switching to X11, I unpinned the apps I wanted to unpin and added the weather applet, and these changes persisted, even after I switched back to Wayland.
I tried to unpin something else, back in Wayland, and this change didn’t persist. So whatever else might be related, I’d say Wayland is part of the issue.
This isn’t quite a solution but it’s a workaround, so I’m marking this topic resolved.