Fix hanging on streams when stdio_client exiting

[LionNatsu]

Triggers a TaskGroup cancel after reaching the finally code block.

Motivation and Context

stdin_writer in the task group hangs when pressing Ctrl+c.
Now we explicitly close the streams blocking.

How Has This Been Tested?

Tested in a new simple no-op test and a real Uvicorn Starlette application.

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