Mike and Clark both make good points. While we technically don't have "socialized medicine" in the US - which is what indigent care is - the indigent care that is available could be much better run.
I believe the problem is that the care is provided by non profit agencies that are poorly run by people who are not adequately trained in the service they provide or in health care administration. They are just people who have a passion about a specific population and try to provide a needed service. A lot of times these agencies duplicate services that other agencies provide, and they compete with each other for limited funding. These agencies also rely on donations from employees in local companies who donate to local charities through United Way and deductions from their pay checks. These agencies also have to hire administrators who know how to jump all the hoops the funding sources require. These administrators generally are paid more than the providers.
I think it would be better to do away with non-profits altogether and develop a more effective system that better manages its resources. That way, more energy can be directed to providing good enough services.... a perfect lead in to the next idea.
In the US, we have come to expect very specialized services from extremely highly qualified providers. We demand the best care available. Most of the time, all we really need is good enough services though.
There is a continuum of affordability and quality of care from indigent to working poor to middle class to wealthy to extremely wealthy. My only response to that is that it's better to accept that life is unfair than to demand that it should be fair.