I got a Tumbleweed badge for this 🙁
FWIW I eventually “fixed” this by restoring the database from backup and recreating the theme files afresh with a new theme name and directory. It seems like a lot more effort than it should be. I figure there was some kind of weird database corruption / conflict somewhere that made it impossible to use widgets that had been used and then removed from the sidebar before being put back.
I could not find a code-based solution, and clearing out references to widgets from the database manually had no effect. I am now terrified of removing and replacing widgets. :/