feat(casting): mount the autoplay watcher and countdown overlay

components/casting/CastAutoplayWatcher.tsx

@@ -0,0 +1,12 @@
+/**
+ * Always-mounted host for the Chromecast autoplay watcher. Renders nothing —
+ * it exists only to run `useCastAutoplay` for the app's lifetime, so autoplay
+ * fires regardless of which screen is open.
+ */
+
+import { useCastAutoplay } from "@/hooks/useCastAutoplay";
+
+export function CastAutoplayWatcher() {
+  useCastAutoplay();
+  return null;
+}