To be hoenst, this is all quite abusive but I'm not sure how to do it in
any better way. I especially dislike the fact that the screen's
bindables are disabled yet the screen itself is allowed to set their
value because they're `LeasedBindable`s.
But I feel better doing this in a central location like this
`ScreenStack` than in the rooms... Otherwise this specific behaviour
would have to be replicated in the multiplayer screen when it goes
through the same refactoring.