In the US it isn't the poor that suffer but the lower-middle class, who are neither poor enough to get excellent health care for free nor wealthy enough to have high-quality private insurance.

The policy problem is that to adequately cover the lower-middle class you would need to substantially raise taxes on the upper-middle class, which pay anomalously low tax rates in the developed world. And the upper-middle class in the US quite enjoys their low tax rates. (While often focused on, the upper class often pays aggregate tax rates in places like California and New York that are more in line with Europe.)