Something I've asked to be done for a long time. Relevant because I've
complained about this on every addition of a new piece of user-local
state: friends, blocks, and now favourite beatmaps.
It's just so messy managing all this inside `APIAccess` next to
everything else, IMO.
Band-aid fix for https://github.com/ppy/osu/issues/32671.
Removes all `BindableList.BindTo` from `load()` methods (except one
editor one which looks safe and is kinda hard to fix without moving
drawable load to a blocking operation).