Make auth output reflect gh auth status

[kostua16]

Summary

• Make auth reflect whether gh auth status --hostname succeeds after login instead of treating token export as enough.
• Add a regression test that mocks gh auth status failure and verifies the action reports auth=false.
• Update the action metadata and README to document the stricter auth contract.

Testing

• Added workflow coverage for the failure case in .github/workflows/test-action.yml.
• Not run (not requested).