Add catch block to prevent silent failure in example TypeScript MCP client (mcp-client-typescript)

[thisistonydang]

Motivation and Context

While following the quickstart guide for the TypeScript MCP Client, the example chat client exited silently after entering a query. This was due to an unhandled error from the Anthropic API (in my case, caused by insufficient credits). Since there was no catch block, the failure was not visible.

This PR adds error handling to surface API issues and prevent silent failures, improving the debugging experience.

PR#499 was also made to update the quickstart docs.

How Has This Been Tested?

I ran the client with the catch block and saw a clear error that I had insufficient credits.

Breaking Changes

N/A

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