fix: align MCP readiness checks

[adity982]

What

Make /healthz aggregate readiness problems and verify that the shared state directory is writable in addition to probing the runtime dataset. Add a focused test for the non-writable-state response.

Why

MCP tools rely on the same agent-owned state as the other transports. Reporting ready while that directory is read-only can admit traffic that later fails during agent actions. The readiness route remains read-only and performs no network access.

The long-term memory database is not opened here because this MCP surface does not own or use that database; the relevant shared-state requirement is the configured state directory.

Validation

• ruff check agents/python/src/agent/mcp_server.py agents/python/tests/test_mcp.py
• Focused direct readiness check: ready state returns 200; mocked non-writable state returns 503 with the expected problem
• git diff --check

Environment note

The repository's locked test environment currently fails during collection before these tests run because google-adk==2.4.0 cannot import google.adk.tools.openapi_tool.auth.credential_exchangers.auto_auth_credential_exchanger. Reinstalling the locked package reproduced the same upstream dependency error.

Closes #62