You're probably right that a true free market health care system is not politically palatable, just like a true free market in any aspect of the economy is not politically palatable. That doesn't mean free markets don't work; it means they're not tried.
free markets are not meant to create a solution for problems whose success is not defined by finding the "fair price" for something
Your definition of a free market is too narrow. Free markets are applicable in any situation where there are scarce resources that need to be allocated. There may not be a "fair" price for health care, but it is certainly a scarce resource that needs to be allocated. There isn't a "fair" price for groceries, computers, etc., but the free market allocates them.
free markets are not meant to create a solution for problems whose success is not defined by finding the "fair price" for something
Your definition of a free market is too narrow. Free markets are applicable in any situation where there are scarce resources that need to be allocated. There may not be a "fair" price for health care, but it is certainly a scarce resource that needs to be allocated. There isn't a "fair" price for groceries, computers, etc., but the free market allocates them.