But they don't.
So, should the hospital, which maintains and provides for an ICU (and not just an ICU - an operating room, an emergency room, beds, a kitchen, heat, air conditioning, a staff of hundreds (at least) ready to care for you at the drop of a hat - should that hospital, which in all probability recieves ZERO of its operating overhead ($100 million/year, probably) from the taxpayers - should it close, or should it send bills?
Which is it?
As it always is, the solution to the problems socialism causes is always more socialism... It never occurs to the bureaucrats that their "solution" is the actual source of the problem they're trying to fix.