fix(scheduler): ensure scheduler starts after setup/config update

Add ensure_started() to SchedulerService as idempotent entry point.
Start scheduler in auth setup run_initialization() after NFO scan.
Sync anime_directory and start scheduler in config update endpoint.
Add unit and endpoint tests for ensure_started() behavior.

tests/unit/test_scheduler_service.py

@@ -559,3 +559,38 @@ class TestStartupRecovery:
                 info_calls = [str(c) for c in mock_logger.info.call_args_list]
                 assert any("next_run" in c for c in info_calls)
 
+
+# ---------------------------------------------------------------------------
+# 12.8  ensure_started() – idempotent startup
+# ---------------------------------------------------------------------------
+
+class TestEnsureStarted:
+    @pytest.mark.asyncio
+    async def test_ensure_started_when_not_running(
+        self, scheduler_service, mock_config_service
+    ):
+        """ensure_started() calls start() when scheduler is not running."""
+        # Mock start method
+        scheduler_service.start = AsyncMock()
+
+        # Call ensure_started
+        await scheduler_service.ensure_started()
+
+        # Verify start() was called exactly once
+        scheduler_service.start.assert_called_once()
+
+    @pytest.mark.asyncio
+    async def test_ensure_started_when_already_running(self, scheduler_service):
+        """ensure_started() returns immediately when already running (idempotent)."""
+        # Set up as already running
+        scheduler_service._is_running = True
+
+        # Mock start method
+        scheduler_service.start = AsyncMock()
+
+        # Call ensure_started
+        await scheduler_service.ensure_started()
+
+        # Verify start() was NOT called
+        scheduler_service.start.assert_not_called()
+