You're conflating private ownership with the organizations nominal financial structure. It has nothing to do with the structure model of the organization and everything to do with resources wasted on TPS reports. This waste has to come from somewhere. Something is necessarily being forgone whether that's profit, reinvestment in the organization or competitive edge that benefits the customer (e.g. lower cost or higher quality for same cost). The same is true for a for profit company, or any other organization.

FWIW the hospital is technically nonprofit as is typical for hospitals. And I assure you, they still have all the wrong incentives despite this.