What works well outside of the US is that the single-payer government run (or at least government mandated) provides a standard level of care, which more often than not includes wellness and life threatening care (and a lot in-between), and one can buy additional coverage in the private market - but the basic standard care is considered a right, and doesn't disappear just because you have some additional coverage.

When you look at other modern nations, all with better performing healthcare systems than the US, there is a mix of arrangements. The common factor is public gov't control of drug prices and procedures. The arrangement of how the administration of the care is allocated varies all across the spectrum, from fully nationalized, to hybrids, to mostly private administration. But I'm inclined to think that public health experts setting cost ceilings by drug & procedure is the critical element.