Fix OBS display and Start button not working

Three issues fixed:

1. Port mismatch: The sidecar reported the OBS port (8080) in the
   ready event but the frontend needs the API port (8081). Now reports
   the API port so WebSocket/REST connects to the right place.

2. Broadcast from wrong thread: Engine init fires state_changed from
   a background thread, but _broadcast_control used get_event_loop()
   which returns the wrong loop. Now captures the uvicorn event loop
   at startup via on_event("startup").

3. Missed ready state: If the engine finishes before the WebSocket
   client connects, the "ready" state_changed was never received.
   Added status polling (GET /api/status) on WebSocket connect that
   retries every 2s while appState is "initializing".

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

backend/main_headless.py

@@ -88,11 +88,16 @@ def main():
         # Create API server wrapping the controller
         api_server = APIServer(controller)
 
-        # Determine actual port (web server may have shifted if port was in use)
-        actual_port = controller.actual_web_port or args.port
+        # OBS display runs on the configured port, API server on port+1
+        obs_port = controller.actual_web_port or args.port
+        api_port = obs_port + 1
 
-        # Print ready event so Tauri can discover the port
-        print(json.dumps({"event": "ready", "port": actual_port}), flush=True)
+        # Print ready event so Tauri can discover the API port
+        print(json.dumps({
+            "event": "ready",
+            "port": api_port,
+            "obs_port": obs_port,
+        }), flush=True)
 
         # Run the API server (blocks)
         import uvicorn
@@ -104,7 +109,7 @@ def main():
         uvicorn.run(
             api_server.app,
             host=args.host,
-            port=actual_port + 1,  # API on port+1, OBS display on the main port
+            port=api_port,
             log_level="error",
             access_log=False,
         )