[infra] Check unclosed/unexpected HTML element in the doc source

[AlexV525]

Describe the problem

https://github.com/flutter/website/pull/12037 contains an unexpected & unclosed <style> element that cause the page rendered incorrectly.

Expected fix

We might consider adding a sanity check to ensure such elements do not exist in the doc source.

On which platform(s) did you experience this issue?

No response

Additional context

No response