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/routes/sync.py

@@ -1,8 +1,8 @@
-from fastapi import APIRouter, Depends
+from fastapi import APIRouter, Depends, HTTPException
 from sqlalchemy import func, select
 from sqlalchemy.orm import Session
 
-from app import quota
+from app import quota, state
 from app.auth import current_user
 from app.db import get_db
 from app.models import Channel, Subscription, User, Video
@@ -79,7 +79,35 @@ def sync_status(
     return {
         "subscriptions": sub_count,
         "channels_total": db.scalar(select(func.count()).select_from(Channel)),
+        "channels_backfilling": db.scalar(
+            select(func.count()).select_from(Channel).where(Channel.backfill_done.is_(False))
+        ),
         "videos_total": db.scalar(select(func.count()).select_from(Video)),
+        "pending_enrich": db.scalar(
+            select(func.count()).select_from(Video).where(Video.enriched_at.is_(None))
+        ),
         "quota_used_today": quota.units_used_today(db),
         "quota_remaining_today": quota.remaining_today(db),
+        "paused": state.is_sync_paused(db),
+        "is_admin": user.role == "admin",
     }
+
+
+@router.post("/pause")
+def pause_sync(
+    user: User = Depends(current_user), db: Session = Depends(get_db)
+) -> dict:
+    if user.role != "admin":
+        raise HTTPException(status_code=403, detail="Admin only")
+    state.set_sync_paused(db, True)
+    return {"paused": True}
+
+
+@router.post("/resume")
+def resume_sync(
+    user: User = Depends(current_user), db: Session = Depends(get_db)
+) -> dict:
+    if user.role != "admin":
+        raise HTTPException(status_code=403, detail="Admin only")
+    state.set_sync_paused(db, False)
+    return {"paused": False}