`CurrentSkinInfo` is used in multiple places expecting thread safety,
while ItemRemoved events are explicitly mentioning they are not thread
safe. As SkinManager itself doesn't have the ability to schedule to the
update thread, I've just moved the logic to `OsuGameBase`. We may want
to move the current skin bindable out of the manager class in the
future to match things like `BeatmapManager`.
Closes https://github.com/ppy/osu/issues/10837.
9a7fdb2b7e