Refactoring needed for new sidebar

Here’s a load of commits to refactor the UI code enough to allow a sidebar to be added (as per #1111 (closed)). These should not change the existing UI behaviour at all (although I may have broken something), but should untangle the code a bit and make it easier to review the changes which add the sidebar. I’ll propose those changes once this has been reviewed. If you want to see them in advance, they’re in this branch.

These are targeting GNOME 41, so should not be merged until we’ve branched. I’ve marked them as a draft accordingly. Apart from that, they are ready to review.

Fixes: #1149 (closed)