Fix non-thread-safe operations in html5

[bdraco]

Proposed change

https://github.com/home-assistant/core/actions/runs/8808425552/job/24177668764?pr=116055

Type of change

• Dependency upgrade
• Bugfix (non-breaking change which fixes an issue)
• New integration (thank you!)
• New feature (which adds functionality to an existing integration)
• Deprecation (breaking change to happen in the future)
• Breaking change (fix/feature causing existing functionality to break)
• Code quality improvements to existing code or addition of tests

Additional information

• This PR fixes or closes issue: fixes #
• This PR is related to issue:
• Link to documentation pull request:

Checklist

• The code change is tested and works locally.
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• Tests have been added to verify that the new code works.

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[bdraco]

These tests could use a lot more refactoring. I did enough to get them to pass but there is more cleanup that should be done in a future PR