Fixed coveo reloading causing query to clear + styling

[lamATnginx]

Proposed changes

• Added a rendered in the DOM atomic-search-box in atomic-search-layout, albeit its hidden. This is because if its not, Coveo has no context or entry point into what the query from the URL is.
• Cleaned up coveo JS code to be more robust and not hard-coded unless absolutely needed.
• Cleaned up styling to be more specific based on comment.

Before:

Screen.Recording.2025-10-16.at.7.11.49.PM.mov

After:

Screen.Recording.2025-10-16.at.7.10.25.PM.mov

Checklist

Before creating a PR, run through this checklist and mark each as complete.

• I have read the CONTRIBUTING document
• If applicable, I have added tests that prove my fix is effective or that my feature works
• If applicable, I have checked that any relevant tests pass after adding my changes
• I have updated any relevant documentation (README.md and CHANGELOG.md)

[github-actions]

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