It seems to me the main problem in the US healthcare is not the question if it's public or not but its large insane costs. The fact that other countries produce the same drugs for a small fraction of the price, sometimes pennies, is very telling of how rotten and inefficient the whole system is.
Imagine the price of milk slowly increasing to $100 per gallon over decades, due to greed, regulations, price cartels and speculation, and nobody noticing to ask why. People would probably being to ration milk only for those who need it the most like mothers who can't breastfeed their babies and there would be political pressure to create public milk banks and milk insurance and other public mechanisms. But all of that political energy would not solve the root causes of the problem.