[dotnet] Avoid potential deadlock when starting new dev tools session

[nvborisenko]

Description

There is a chance to get a deadlock when starting new cdp session.

Motivation and Context

Fixes #11703

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)

Checklist

• I have read the contributing document.
• My change requires a change to the documentation.
• I have updated the documentation accordingly.
• I have added tests to cover my changes.
• All new and existing tests passed.

I have tested it locally, the issue is not reproducible anymore.

[titusfortner]

Does #12614 affect this?

[nvborisenko]

Should not, it's still actionable.