Make all objects stateless.

Specifically:
1. Instead of `player.team_color`, pass `team_color` in state each turn.
2. `GameManager` should represent just a single game (no init game, reset state, etc).

Start by tracing the player objects and GameManager, especially the player.team_color access and any initialization or reset-state paths. Determine how turn state is passed today; done means team_color is supplied in each turn's state and GameManager represents only one game without game initialization or reset responsibilities.

Written by the indexing model from the issue text.