Extracts checkRoomStatus into shared jackbox-api.js with proper
User-Agent header (bare fetch was silently rejected by Jackbox API)
and always-on error logging (previously gated behind DEBUG flag).
Splits room-start detection (room-monitor.js) from audience-based
player counting (player-count-checker.js) to eliminate circular
dependency and allow immediate game.started detection. Room monitor
now polls immediately instead of waiting 10 seconds for first check.
Made-with: Cursor
