fix(game): trigger animations and errors display in the API service instead of the layout

It is more coherent to let services manage this independently of whether the elements are ready in the DOM or not.

We also reintroduce the child route for the game because:

• It makes sense.
• It is the easiest and cleanest way to prevent aot-popup to be destroyed and re-created when going to the play page. This caused the first transition popup to be destroyed while it was displayed.