I think this is a problem orthogonal to private versus public healthcare. I think it is a problem relating to what health insurance is today. Pretty much all hospitals get all their indirectly from patients through insurance agencies. And this is for all services from checkups to bloodwork to scheduled surgeries to emergencies.

Since the insurance companies are the middle man, the cost structures become less directly correlated between patient behavior. Demand is inelastic, particularly because insurance smooths it out. The hopsital needs so much money, so why not inflate the price of advil over here, even though it doesn't directly correlate to their costs.