A better idiom for your use case might be ‘carry water’. Saying Tim Cook carries water for the CCP would be to say he does the work of the CCP, equivalent to to

A better idiom for your use case might be ‘carry water’. Saying Tim Cook carries water for the CCP would be to say he does the work of the CCP, equivalent to towing a load. However, I think this is a stronger phrase than what the original commenter intended.