Fix async loading bugs and add test results

Critical Fixes:
- Fix async context manager usage in fastapi_app.py (async for -> async with)
- Add broadcast() method to WebSocketService
- Initialize BackgroundLoaderService properly in lifespan function

Testing:
- Execute manual testing (Tests 1, 5, 8, 9)
- Create comprehensive test results document
- Verify API endpoints return 202 Accepted
- Confirm database persistence works
- Validate startup incomplete series check

Test Results:
- Response time: 61ms (target: < 500ms) ✅
- 4 series found with missing data on startup
- Database fields properly persisted
- All critical bugs fixed

Files:
- check_db.py: Database inspection utility
- docs/MANUAL_TESTING_RESULTS.md: Comprehensive test results
- src/server/fastapi_app.py: Fixed async context manager, initialized BackgroundLoaderService
- src/server/services/websocket_service.py: Added broadcast() method

check_db.py

@@ -0,0 +1,35 @@
+"""Simple database check script for manual testing."""
+import asyncio
+
+from sqlalchemy import select
+
+from src.server.database.connection import get_db_session, init_db
+from src.server.database.models import AnimeSeries
+
+
+async def check_series():
+    """Check series in database."""
+    await init_db()
+    
+    async with get_db_session() as session:
+        result = await session.execute(select(AnimeSeries))
+        series_list = result.scalars().all()
+        
+        print(f'\n=== Database Series Check ===')
+        print(f'Total series: {len(series_list)}')
+        print()
+        
+        for s in series_list:
+            status = getattr(s, 'loading_status', 'no field')
+            episodes = getattr(s, 'episodes_loaded', 'N/A')
+            nfo = getattr(s, 'nfo_loaded', 'N/A')
+            logo = getattr(s, 'logo_loaded', 'N/A')
+            images = getattr(s, 'images_loaded', 'N/A')
+            
+            print(f'{s.name} ({s.key}):')
+            print(f'  Status: {status}')
+            print(f'  Episodes: {episodes}, NFO: {nfo}, Logo: {logo}, Images: {images}')
+            print()
+
+if __name__ == '__main__':
+    asyncio.run(check_series())