Aman, you started by asking how the market can provide necessities. Surely it is relevant to the question that when the market is allowed to work it does, in fact, provide things even more necessary to life than health care: food, water, shelter, and so on in significant amounts at reasonable prices.
It doesn’t matter at all that some of the producers are highly trained professionals. This is true in the production of some aspect of almost every good in the division of labor. If the wage is high for highly trained professionals, then it gives people monetary incentive to enter these fields, which draws more people into them, which moderates their wage. In the market, the efficient number of people are drawn into every occupation.
The (subjective) value a man in the hole places on the ladder is very high, but the price of the ladder in a market economy is low. For every good, some people will place high subjective value on it, others a moderate amount, and others even less. But the price is the same for everyone. It must be low enough to clear to the market given past production based on what entrepreneurs anticipated the profit of production would be.
If people think they will be in a position to be ripped of because of unfortunate circumstances, then they can contract with suppliers before they get into trouble. They buy a rope ladder and carry it with them.
Entrepreneurs will even accommodate people in their attempts to avoid having to make trades under duress. Insurance companies permit people to pool their risk..
You say, “Isn’t this what is happening in fact in our society as the free market allowed to be in charge of delivering healthcare to people?” In our society, there is massive and increasing government intervention. Rising prices in health care come from the government increasing demand through the spiraling federal expenditures and the special privileges given to third party payments and restricting supply through licensing and other restrictions.
There are lots of resources. For example: