Ensure all template errors are caught and the websocket api reports them

[bdraco]

Supports #41703

Proposed change

Ensure all template errors are caught and the websocket api reports them

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

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• This PR fixes or closes issue: fixes #
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[bdraco]

Should we tag this for 0.116.x as it likely fixes the reports of templates that stop rendering and never resume?

It looks like there was trouble figuring out why the template was failing in #40221