Prevent stdio connection hang for missing server path.

[TimChild]

• Ensure that read/write streams are always closed (even if it is the process that ends first)
• Check whether the stdio process ended prematurely
• Raise an error if that was the case

Motivation and Context

The stdio connection was hanging indefinitely for a bad server file path (e.g. command uv, args run non-existing-file.py).
I think because the read/write streams were not being closed properly in that case.

This makes it so that errors due to the underlying mcp server stopping early raises a ProcessTerminatedEarlyError.

How Has This Been Tested?

Added a test (including catching if it does hang for more than 1s)

Breaking Changes

Should not be a breaking change.

Types of changes

• [ x ] 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

• [ x ] I have read the MCP Documentation
• [ x ] My code follows the repository's style guidelines
• [ x ] New and existing tests pass locally
• [ x ] I have added appropriate error handling
• [ x ] I have added or updated documentation as needed -- Did not update docs (but error is self-explanatory)

Additional context