“They’ll just put out list prices that are five to six times as high as what they’ll actually get.”
That wouldn’t work with stations selling gas or grocers selling bread. Why do you think it would work for hospitals? Public pricing of a medical service would bring down the cost as hospitals compete for patients.
Because unlike at supermarkets or gas stations almost no one pays the "standard" price - their insurance company pays whatever rate they've negotiated with the hospital.
Notice that the EO doesn't require insurance companies to post the rates they've negotiated with the providers.