perf: remove redundant readiness query

[Linxiushen]

Summary

• rely on the engine's configured pool_pre_ping during readiness connection checkout
• remove the handler's redundant second SELECT 1 database round-trip
• tighten the readiness test double so it fails if the endpoint attempts to execute a query

The endpoint still acquires a database connection and preserves the existing 503 response for connection or pre-ping failures.

Testing

• ruff check openhands/automation/app.py tests/test_health.py
• ruff format --check openhands/automation/app.py tests/test_health.py
• full endpoint tests were not available locally because LiteLLM 1.93's Windows build requires the MSVC linker; upstream Linux CI should run them

Fixes #287

Disclosure: This change was prepared with AI assistance. I reviewed the implementation and ran the checks listed above.