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

frontend/src/components/Feed.tsx

@@ -1,5 +1,5 @@
 import { useEffect, useRef, useState } from "react";
-import { useInfiniteQuery, useQueryClient } from "@tanstack/react-query";
+import { useInfiniteQuery, useQuery, useQueryClient } from "@tanstack/react-query";
 import { api, type FeedFilters, type Video } from "../lib/api";
 import { toast } from "../lib/toast";
 import VideoCard from "./VideoCard";
@@ -42,6 +42,12 @@ export default function Feed({
 
   useEffect(() => setOverrides({}), [filters]);
 
+  const countQuery = useQuery({
+    queryKey: ["feed-count", filters],
+    queryFn: () => api.feedCount(filters),
+    staleTime: 30_000,
+  });
+
   const sentinel = useRef<HTMLDivElement | null>(null);
   const { hasNextPage, isFetchingNextPage, fetchNextPage } = query;
   useEffect(() => {
@@ -87,6 +93,11 @@ export default function Feed({
 
   return (
     <div className="p-4">
+      <div className="pb-3 text-sm text-muted">
+        {countQuery.data
+          ? `${countQuery.data.count.toLocaleString()} video${countQuery.data.count === 1 ? "" : "s"}`
+          : " "}
+      </div>
       {view === "grid" ? (
         <div className="grid gap-4 grid-cols-[repeat(auto-fill,minmax(260px,1fr))]">
           {items.map((v) => (