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/scheduler.py

@@ -6,6 +6,7 @@ from apscheduler.schedulers.background import BackgroundScheduler
 
 from app.config import settings
 from app.db import SessionLocal
+from app.state import is_sync_paused
 from app.sync.autotag import run_autotag_all
 from app.sync.runner import (
     run_deep_backfill,
@@ -24,6 +25,9 @@ _scheduler: BackgroundScheduler | None = None
 def _job(name: str, fn) -> None:
     db = SessionLocal()
     try:
+        if is_sync_paused(db):
+            logger.info("job %s skipped (sync paused)", name)
+            return
         result = fn(db)
         logger.info("job %s -> %s", name, result)
     except Exception: