This is the crux of the debate. I tend to believe that if private insurance is allowed to stand, the healthcare system would eventually come to mirror the education system with heavily bifurcated quality streams. Imagine how much better the public school system would be if private/charter schools were outlawed. Capital would be constrained within the system instead of following the rich through the escape hatch of private schools/healthcare.
The healthcare systems of the UK, Canada, Scandinavia, and other Western European nations benefit from being formed in a time when public spending wasn't constantly under attack, and in countries with far more egalitarian leanings than modern America. Even now though, the NHS is under constant attack from the Tories