I think integrated systems could help, where the financial and operational sides of healthcare are combined (e.g. Group Health in Seattle, or Kaiser). The key thing is to remove the conflict between the payer and operator -- it's just madness that we have a system where one party has 100% of the incentive to control costs, and the other is completely responsible for outcomes, cost be damned.
My girlfriend is a doctor here in Europe (though she has worked in Mexico). As such she has worked with private and public systems (generally in Europe, you are automatically part of the public system. You can pay private yourself and generally have to wait less).
Anyway, both her and her boss pointed out that if you have anything serious, go to the public system. Why? Because insurance companies always want to pay the least for the cheapest drugs / treatment. The doctors don't enjoy working for the private, because too much of it is trying to justify using the more expensive treatment to the insurance companies.
At the end of the day the public healthcare system exists to help people get better. The private healthcare system exists to make money.
Administrative overhead of government vs private insurance: http://healthcarereform.procon.org/view.resource.php?resourc...
http://www.commonwealthfund.org/~/media/Files/Publications/I...
Essentially, we excel in a couple very specific things, like survivorship of breast cancer. We're middle of the pack in some things, like general cancer survivorship. And we're the worst by quite a bit in others, like chronic diseases. Basically, I would say we're not getting our $'s worth.
Here's some Commonwealth Fund info if you don't want to trust some random white paper.
http://en.wikipedia.org/wiki/Commonwealth_Fund
Oh, and if you want something from an organization with a known bias: http://www.rand.org/content/dam/rand/pubs/working_papers/201...
Key abstract quote: "But one key finding emerges – the US ranks poorly on all indicators with the exception of self-reported subjective health status."
Americans actually display stunning recalcitrance towards this fact, and as a result we have an incredibly polarized debate which has led to a bastardized and amalgamated system comprised of several other, and often conflicting, constructs.