The free market would never build this without the government paying, because most patients can't exactly afford to pay for their share of the construction cost

The free market would never build this without the government paying, because most patients can't exactly afford to pay for their share of the construction costs. For-profit hospitals are incentivized to turn away patients showing concerning symptoms to prevent them from infecting their paying customers. There are many lessons that can be drawn from this, but "privatize everything" is not one of them.