Resolved, it still doesn't use a select list, but the problem of headlines disappearing when trees are renamed is fixed.
The Base headlines path field that comes with the Headlines Component currently takes a path, this is not so good for several reasons:
Consistency, select lists are used elsewhere
Renaming — when a /foo-bar/ documents is renames to /foo/ the headlines components listing child documents of /foo-bar/ will break.
It should use a select list and it should store the document ID rather than the path.