Decompose GameScreen to match other screens' modularity

[francinv]

GameScreen is 448 lines with 5 files. SetupScreen is 488 lines with 11 files. The split-module tactic was applied unevenly.

GameScreen still owns:

• Top-bar UI layout
• Tap-selection state machine
• Move execution + turn logic
• Promotion flow
• Heartbeat scheduling
• Network callback handling

What to do:

• Extract UI construction into a dedicated overlay/UI class
• Extract tap-selection + move execution into a game input handler (like SetupBoardInputHandler)
• Extract promotion flow into its own class
• GameScreen should coordinate, not implement

Ref: SetupScreen's decomposition (renderer, input handler, flow controller, widgets, codec, mapper) is the pattern to follow.