cloud_api_client: Make `send_authenticated_json_request` public (#60562)

This PR makes the `send_authenticated_json_request` method on the
`CloudApiClient` public.

This way we can use it to make requests by external callers.

The `build_request` method was also inlined into
`send_authenticated_request` to make the contract simpler.

Release Notes:

- N/A
f9c994796aMarshall Bowers committed on 7/7/2026, 11:00:12 PM· committed by GitHubparentd12b980
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