That's not entirely true and a misstatement of how these work, take Blue Cross of Michigan. It is a nonprofit, it's board is controlled by the members, essentially they are the share holders, they have a interest in lower prices and an efficient operation.
How the board gets approved to be elected is a bit self selected, but in theory it should be controlled by people that have incentives, the same as a public corporation.
Now this is an insurance company and not a provider network,some provider networks are more self owned.