feat: sync status indicator, admin pause/resume, filtered video count

- Header shows a live sync status (total videos + channels still backfilling, or
  "paused"), polled every 30s
- Admins can pause/resume the background sync; a paused flag in app_state makes
  scheduled jobs skip (migration 0006)
- GET /api/feed/count returns the number of videos matching the current filters;
  shared filter builder keeps it in sync with /api/feed; shown above the feed
- /api/sync/status reports backfill progress, pending enrichment and paused state

backend/app/state.py

@@ -0,0 +1,24 @@
+"""Global admin-controlled app state (e.g. pausing background sync)."""
+from sqlalchemy.orm import Session
+
+from app.models import AppState
+
+
+def _row(db: Session) -> AppState:
+    row = db.get(AppState, 1)
+    if row is None:
+        row = AppState(id=1, sync_paused=False)
+        db.add(row)
+        db.commit()
+    return row
+
+
+def is_sync_paused(db: Session) -> bool:
+    return _row(db).sync_paused
+
+
+def set_sync_paused(db: Session, paused: bool) -> None:
+    row = _row(db)
+    row.sync_paused = paused
+    db.add(row)
+    db.commit()