I reflexively say we had a cook, not hired one. Ownership is embedded in the verb.
Reflections on Having a Servant
Bernie Bleske

I’m not certain your distinction between “having" and “hiring" works to reveal anything of the structure of the relationship between a client and a service provider. People tend to say they “have" a lawyer, a doctor, or an accountant, and not that they “hired" one. The use of the word “client" in these relationships suggests that one is the inferior the relationship.