Fix typo & Remove a useless instance and finalize the injecting property

[echo-youn]

Fix typo deafault.. into default.
Remove a useless instance and finalize the injecting property.

I remain wrong class for backward capabilities as deprecated

Motivation and Context

Correcting a typo to maintain consistency and avoid any potential misunderstanding.
Replacing field with constructor injection and marked the property as final for better immutability and testability.

How Has This Been Tested?

I test initiate server sync and async both. It was run correctly.

Breaking Changes

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