cloud_llm_client: Remove unused code (#37799)

This PR removes some unused code from the `cloud_llm_client`.

This was only used on the server, so we can move it there.

Release Notes:

- N/A
e4de26e5dcMarshall Bowers committed on 9/8/2025, 8:42:42 PM· committed by GitHubparent7091c70
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