Implement Missing #checkGhAuth Method in HealthService

A code review revealed that the HealthService.healthcheck method calls a private method, #checkGhAuth(), which does not exist in the file. This is a critical bug that causes the health check to fail and prevents the server from starting.

This ticket covers the implementation of the missing authentication check.

Acceptance Criteria

1. A new private method, #checkGhAuth(), is implemented in HealthService.mjs.
2. The method executes the gh auth status command.
3. It correctly parses the output of the command to determine if the user is authenticated with github.com.
4. It returns an object with the shape { authenticated: boolean, error?: string }.
   • authenticated should be true if login is successful, false otherwise.
   • error should contain an informative message if authentication fails.