"Government" is a human creation.   There are no natural limitations to government, because government isn't natural.  Therefore, government is whatever we want it to be.   There are no more restrictions to using government (a way of collectively organizing, cooperating and achieving goals) to provide healthcare than there are to using government for the provision of education or police or fire departments or mail service, or anything else humans choose to use government for.  We made it up, and we get to use it any way we want.

The problem is us.   We don't use government well.