We don't need them to. 62% of hospitals in the US are non-profit, and an additional 20% are run by the government. Only 18% are for-profit. In some states, like New York and Minnesota, there are no for-profit hospitals.
Example: https://www.propublica.org/article/methodist-le-bonheur-heal...
Some people in the medical profession may be willing to work for less money for a charitable cause, but not enough to run entire hospital systems.