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>

src/lib/stores/backend.svelte.ts

@@ -54,6 +54,35 @@ async function apiFetch(path: string, options?: RequestInit): Promise<Response>
   return fetch(url, { ...options, headers });
 }
 
+// ── Status polling ──────────────────────────────────────────────────
+
+let statusPollTimer: ReturnType<typeof setTimeout> | null = null;
+
+async function pollStatus() {
+  try {
+    const resp = await fetch(apiUrl("/api/status"));
+    if (resp.ok) {
+      const data = await resp.json();
+      if (data.state) {
+        state.appState = data.state as AppState;
+      }
+      if (data.engine_device) {
+        state.deviceInfo = data.engine_device;
+      }
+      if (data.version) {
+        state.version = data.version;
+      }
+    }
+  } catch {
+    // API not ready yet, will retry
+  }
+
+  // Keep polling every 2s while still initializing
+  if (state.appState === "initializing" && state.connectionState === "connected") {
+    statusPollTimer = setTimeout(pollStatus, 2000);
+  }
+}
+
 // ── WebSocket management ─────────────────────────────────────────────
 
 function connectWebSocket() {
@@ -80,6 +109,9 @@ function _openSocket() {
         clearTimeout(reconnectTimer);
         reconnectTimer = null;
       }
+      // Poll status to catch engine ready state that may have been
+      // missed (engine can finish before WebSocket connects)
+      pollStatus();
     };
 
     ws.onmessage = (event) => {
@@ -132,6 +164,10 @@ function _scheduleReconnect() {
 }
 
 function disconnect() {
+  if (statusPollTimer) {
+    clearTimeout(statusPollTimer);
+    statusPollTimer = null;
+  }
   if (reconnectTimer) {
     clearTimeout(reconnectTimer);
     reconnectTimer = null;