That exists in India in the form of societies like Lions Club, Rotary Club etc. However, the number of hospitals that are supported by these charitable societies are understandably smaller in comparison to the number of private hospitals.
So when we take a look at societies that actually do it this way, such as India today or the USA before 1986, why do we find hospitals dumping patients who can't pay?
If they were able to bill insurance, or the person, later, then the fund would be replenished.
So it is possible to say "yes" and not have government involved.