With `ReplayPlayer` now consuming the `LeaderboardManager`'s global
state, flows such as presenting a score need to set the global state up
correctly to avoid accidentally showing a leaderboard from a completely
different score.
This also incidentally closes https://github.com/ppy/osu/issues/27609.
This one probably doesn't matter as much because it's used as a single
instance in `TaikoPlayfield` (so its lifetime should end around the same
time as the `HealthProcessor`).
Beyond isolation, this stuff is also used to test (de-)serialisation.
Much-of-a-muchness in this case since it isn't too thoroughly tested,
but best do it anyway.