No one is making the claim that the cost of service would be $0. Quite obviously.

What you can't be sure of is the real cost in terms of resources, and how that would be accounted and paid for. The price of your childrens' stay, to you personally, could be far less than what you originally paid personally if healthcare was not reliant on private subsidy. You would also likely be subsidizing the costs of other children staying in the hospital through public subsidy using your tax dollars. I reckon that last point is where a lot of people in the US get hung up.