test(auto-updater): add debug logging to understand CI failure

[baleen37]

This PR adds comprehensive debug logging to test #6 to understand why timestamp file creation fails in CI but passes locally.

[coderabbitai]

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✨ Finishing touches

🧪 Generate unit tests (beta)

• Create PR with unit tests
• Post copyable unit tests in a comment
• Commit unit tests in branch debug/test-timestamp-env

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