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/alembic/versions/0006_app_state.py

@@ -0,0 +1,28 @@
+"""app_state (admin-controlled global flags)
+
+Revision ID: 0006_app_state
+Revises: 0005_shorts_probed
+Create Date: 2026-06-11
+"""
+from typing import Sequence, Union
+
+from alembic import op
+import sqlalchemy as sa
+
+revision: str = "0006_app_state"
+down_revision: Union[str, None] = "0005_shorts_probed"
+branch_labels: Union[str, Sequence[str], None] = None
+depends_on: Union[str, Sequence[str], None] = None
+
+
+def upgrade() -> None:
+    op.create_table(
+        "app_state",
+        sa.Column("id", sa.Integer(), primary_key=True),
+        sa.Column("sync_paused", sa.Boolean(), nullable=False, server_default="false"),
+    )
+    op.execute("INSERT INTO app_state (id, sync_paused) VALUES (1, false)")
+
+
+def downgrade() -> None:
+    op.drop_table("app_state")