Fixes #753

Motivation and Context

This PR adds a drain phase between the main loop exit and transport.close(). It also changes response send tasks from fire-and-forget tokio::spawn to a tracked JoinSet, so we can wait for sends already dispatched by the main loop before draining any remaining handler responses from the channel. A configurable timeout (5s for Closed, 2s for Cancelled) prevents hanging on stuck handlers. The drain is skipped entirely for JoinError quit reasons.

How Has This Been Tested?

Added new integration tests.

Breaking Changes

None. The drain phase is purely additive.

Types of changes

• Bug fix (non-breaking change which fixes an issue)
• New feature (non-breaking change which adds functionality)
• Breaking change (fix or feature that would cause existing functionality to change)
• Documentation update

Checklist

• I have read the MCP Documentation
• My code follows the repository's style guidelines
• New and existing tests pass locally
• I have added appropriate error handling
• I have added or updated documentation as needed

Additional context