I've taken part of the healthcare systems in the US, Norway, and now Czech Republic. Norway had the highest level of quality in terms of their healthcare. Czech Republic is by far the most cost efficient one (I buy commercial insurance that costs $~750 a year and gives coverage everywhere in the EU). The US had the most complicated and expensive system.
Seeing as the US has the most privatized system out of the three, I'd say that "getting the government out of the equation" won't necessarily make it more affordable.
Even if it did make it more affordable to completely privatize the system, do you really want to "shop around" for healthcare services? When I have an emergency I just want to go to a hospital as quickly as possible. I don't think anyone wants to be comparing prices/packages while holding a severed finger in their hand.
The US may be relatively privatized, but it is a system where hospitals will bill patients a certain way because they expect most patients to be using their insurance plans to pay for it.
With many doctors and hospitals you can barter over a procedure by offering cash. It can take a very long time for a hospital to collect on money owed by the insurance companies for the procedure, so cash is always preferable when a hospital is for-profit.
The american healthcare system has a middle man who is disinterested in the prices of health services because they can simply raise premiums. Saving the consumer more cost in premiums is not the prime motivation of insurance companies.
When a hospital bills the insurance company $300 for a bottle of saline solution (found on Amazon.com for $2) the insurance company isn't going to do the same kind of haggling that you will if you were being handed that bill for the cash in your pocket.
In response to the ER visit comment: If we were living in a world where what I said above was true, and that the patient was dealing directly with hospitals without any insurance company involvement, chances are you will end up at a hospital which also is forced to compete on price. In a true emergency, a few dollars difference isn't going to matter.