Fix stdio_client kill process after timeout

[cristipufu]

Motivation and Context

Be consistent across platforms (Windows, Linux, etc). Some MCP Servers don't react to SIGTERM, so stdio_client will hang indefinitely when exiting the context manager.

Reference: https://github.com/modelcontextprotocol/python-sdk/pull/372

How Has This Been Tested?

Yes

Breaking Changes

No

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

[surya-prakash-susarla]

Slightly orthogonal but this does not handle child process spawns which can cause the parent to leave ghost processes which are being re-parented to the root process. Addressed it here: https://github.com/modelcontextprotocol/python-sdk/pull/850

[cristipufu]

@felixweinberger rebased

[felixweinberger]

Thank you @cristipufu for your response and again for submitting this PR!

I spent yesterday and today trying to unify all the different approaches and fixes we have pending in this process termination space at the moment, as there are several interrelated fixes that either conflict or depend on each other - specifically #555, #729, #765, and #850.

I've added your change to #1044 as a draft with you as a co-author + added extensive regression testing. Would you be OK with consolidating this change into #1044 for the comprehensive testing & process handling introduced there?

[cristipufu]

@felixweinberger yes, of course, thanks!