Reported at https://osu.ppy.sh/comments/3681620, with appropriate levels
of rage bait (DID ANYONE TEST THIS?!?!?!?!?!?!?!?!?!111!!)
Reasoning for this is that without this, users' skin names can be
dropped after an external edit because they're never persisted anywhere
outside of realm.
The only other choice I see is to stop re-populating skin metadata from
the `.ini` upon completing an external edit, which is very doable but
seems worse than this. Dunno.
Now, nothing touches the configuration apart from `OsuGame`, making
everything else flow better and avoid weird cyclic set bugs.
Closes https://github.com/ppy/osu/issues/20234.