docs: add nested try/finally cleanup to root TypeScript README example (#2726)

[teddiesloco]

Summary

• Fixes Root TypeScript README example omits Stagehand and Browserbase cleanup #2726
• Updates the canonical TypeScript example in README.md to use nested try/finally blocks, ensuring stagehand.close() and browser.close() are called on both success and error.
• Aligns root quickstart with Python and Go READMEs and prevents billable Browserbase session leaks.

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Summary by cubic

Fixes #2726 by updating the root TypeScript README example to use nested try/finally blocks, so stagehand.close() and browser.close() run on both success and error. This aligns the quickstart with the Python and Go READMEs and prevents billable Browserbase session leaks.

^{Written for commit e7bab24. Summary will update on new commits.}

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Latest commit: e7bab24

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[github-actions]

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