refactor: extract mode controllers from GameApp (plan 42)

Split game-app.ts (1,532 lines) into four focused files with no
behavior changes:

- loading-controller.ts: owns WorldLoadingState and loading token
  machinery (beginLoading, isLoadingTokenActive, applyLoadingProgress,
  failLoading, etc.) plus the loading-screen tick UI
- menu-controller.ts: owns MenuState, menu tick, all action dispatch,
  and world/server management helpers (syncMenuWorlds, syncSavedServers,
  createWorld, deleteWorld, saveServer, deleteServer)
- play-controller.ts: owns chat/inventory/pause state,
  firstPersonSwingRemaining, debug-memory helpers, the fixed-step
  gameplay tick (updateGame), input routing, HUD build, and overlay text
- game-app.ts: reduced to 719 lines; retains connections, appMode,
  timing, settings, joinWorld/joinServer, completeWorldJoinLoading,
  registerConnectionEventHandlers, renderFrame, and the main tick
  dispatcher

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

Author: r1tsuu