Doctors and nurses are rational humans who maximize their own utility, just as patients try to maximize their utility by choosing to demand services from the former. Maybe the patient could try paying the workers of a hospital in bricks, but moving cash around is easier.
Is it hard to understand that healthcare exists as a service, that is produced and demanded; and the way the finite resources is allocated would operate best under free competition, which always maximized the benefit of the customer on the long run due to competition enabling price deflation?
There is no such thing as 'free' in this world of ours. Deal with it. Healthcare does not rain down from the sky, nor does it grow in trees. It is a service like many others. People consume services to maximize their utility. Such consumption will be limited by the budget restrictions of individuals, just as the supply of the service will be limited due to doctors only being able to 24 hours a day in theory. So we face a problem where a good now has a value.
And the value of the good forms on the markets, according to the rationale of individuals who deem the good according to their marginal utilities.
The price comes to being; a price must be paid for the good.
Questions?