Add root_path support to properly route subpaths for SSE and streamable-http endpoints when behind Nginx or other HTTP reverse proxies

[LiangYang666]

This PR adds support for root_path in subpath routing of SSE andstreamable-http endpoints when using Nginx or other HTTP middleware as a reverse proxy.

Motivation and Context

When deploying applications behind a reverse proxy (e.g., Nginx), it's common to route requests via subpaths like /server1/ or /server2/ . Without proper handling of these subpaths, SSE and streamable-http endpoints may fail to receive correctly routed requests.

This issue was originally reported in #242

Similar functionality exists in other web applications:

• base_url in jupyter
• base_url in filebrowser
• root_url in grafana

How Has This Been Tested?

Yes, this has been tested and running stably in production environments for several days. Both sse and streamable-http endpoints have been verified to work correctly under subpaths behind Nginx.

Breaking Changes

No breaking changes are introduced. Users do not need to update their code or configuration files. This is an optional enhancement that activates only when root_path is provided.

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

This change ensures that the application respects the root_path setting for streaming endpoints by utilizing Uvicorn’s built-in support for it. It does not introduce any custom routing logic but instead makes sure that streaming responses behave consistently with regular HTTP responses under subpath deployments.