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

docs/instructions.md

@@ -161,6 +161,7 @@ Successfully implemented asynchronous series data loading with background proces
 **Manual Testing Guide:**
 
 A comprehensive manual testing guide has been created at `docs/MANUAL_TESTING_ASYNC_LOADING.md` with:
+
 - 10 detailed test scenarios covering all functionality
 - Step-by-step instructions with expected results
 - Troubleshooting section for common issues
@@ -168,6 +169,7 @@ A comprehensive manual testing guide has been created at `docs/MANUAL_TESTING_AS
 - Test results template
 
 **How to Test:**
+
 1. Start the server: `conda run -n AniWorld python -m uvicorn src.server.fastapi_app:app --host 127.0.0.1 --port 8000 --reload`
 2. Follow the test scenarios in `docs/MANUAL_TESTING_ASYNC_LOADING.md`
 3. Verify all 10 test scenarios pass