Non American here.
They already know and/or could trivially find out. There's no more barrier to them finding out now then there would be if there was some new, nominally independent government entity involved. That is to say a very thin wall preventing what you describe.
I sometimes get the feeling we oftentimes allow for market absurdities to supposedly prevent the government from doing something that it regularly already does today, just as long as there's a 'business' somewhere in the chain.
Americans seem willing to pay more to private business and thus actually part with more of their money than they otherwise would, as long as on paper they're not giving the money to 'the government' see i.e. healthcare.