Fix automatic intro skip not triggering in multiplayer

[intkr]

Closes #36783 and #37906.

Ideally the clock is reset on onGameplayStarted, then the intro is automatically skipped on RequestIntroSkip (from SkipOverlay.SkipWhenReady).
However, if at least one other player stays in PlayerLoader, the players who enter the loaded state first are able to click the skip overlay before the gameplay starts.
This causes the clock to skip the intro first, then reset back to zero as the gameplay starts.

[peppy]

Please add failing test coverage and inline comments explaining what is going on.

[intkr]

Test passes on PR / fails on master, but currently has two problems.

1. The map used in TestBeatmap does not have a skippable intro, and the test will fail from trying to click a nonexistent skip overlay button. I have confirmed the test works by changing the beatmap string locally, but that isn't included in the PR. In the event changing the beatmap is allowed, I think this map with a 6 second intro may be a good alternative.

2. ServerAPIRoom, the Room used for initializing MultiplayerPlayer in the test scene, doesn't update after the room is created. I was not able to enable auto skip from just calling ChangeSettings().
   The two methods that made it possible was to either create a Room with AutoSkip enabled for JoinRoom(), or use ClientAPIRoom (which does get updated from changing settings) when initializing gameplay instead. PR currently uses the latter but I'm not sure which one I should go with.

Converting to draft until both are resolved.