My argument is not that government is inefficient and so it shouldn't be used, my argument is that despite inefficiencies it's appropriate to use in certain cases because the alternatives are worse.
I think with regards to healthcare, the private insurance market has proven to be insanely inefficient. We have a patchwork of insurance laws across states, profit motivated insurance companies, and just overall opacity at every level of the system. All this and if you look at more centralized government managed systems, overall outcomes seem no worse and in some cases better. It's hard to measure quality in healthcare, as I mentioned above, but by many of those metrics the US system isn't doing radically better than, say, European systems which are largely single payer government organized or close to it. All that despite paying as much as 30% more.
There are lots of questions and issues in healthcare, it's a really complicated question. One thing that's pretty clear is that as much as 50% of your entire lifetime healthcare spend will be spent in your last 6 months of life. At that stage how can a private market work? "Hey, I'm dying let me shop around for the best ICU"?
All of the alternatives involve tradeoffs, usually in autonomy in deciding care (single payer can become a huge HMO) and difficulty in ensuring appropriate supply (waiting months for a CT scan you actually needed yesterday). I believe there are ways to solve those even with a centralized system, and I'm fairly convinced that our private market has done a poor job serving the majority of patients (unless you're fairly wealthy, and even then the structural inefficiencies of a fragmented system get in the way).
Re. private audit and review companies: It turns out a lot of docs are on Yelp and other similar services. I'll admit to looking at them when I look for a provider. But those systems have really perverse effects as well. People usually don't know what the is best thing for them to do medically, but they love it when the doc agrees with them. As a result a doc is pushed toward being more agreeable, prescribing what the patient wants instead of what they need.
Patient says, "I want antibiotics". Doc says, "but it's likely viral". Patient says, "I want antibiotics". The doc who caves gets better reviews.
Good medicine is hard to evaluate, and a patient is poorly equipped to evaluate relative quality, but he/she is also poorly informed as to what is a good price. You could say, "let's get rid of opacity." But one of the drivers for opacity is the private insurance system. No insurer wants what they're paying for X procedure to be known by their competitors, that complicates their negotiating position.
You could force regulation that would open all this up, but then aren't you just growing government involvement in healthcare regulation?
With so many asymmetries and so much opacity it's hard to see how a private system does better. That said, it's worthy of discussion and a complicated topic, so I really enjoy learning of other opinions.