refactor: improve CLI type safety and log level consistency

[steviec]

The CLI unnecessarily ignored some of the typescript SDK's types, which made it less type-safe and harder to maintain. The log levels were also hardcoded as strings instead of pulled from the types exposed by the SDK, which caused log levels to be inconsistent across the codebase.

Motivation and Context

I was working on a PR for a new CLI feature and noticed that a lot of the types were "any", which was a smell to me knowing that the typescript SDK uses strong typing. Instead of including this refactor in that PR, I thought it would be better to pull it out as a standalone PR.

How Has This Been Tested?

Tested locally using the CLI against my local MCP server, and all linting/testing run for the repo.

Breaking Changes

No, it should not affect anyone.

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

None.