The government is indeed subject to the law, much more stringently than insurance companies in fact. Try doing something nominally illegal in a government agency.
The government can pass laws, of course.. but the insurance industry can hire lobbyists. Seems like an awful uphill battle to beat those guys, based on the last 60 years of healthcare reform history. Harry and Louise can be pretty influential.
Medicare's dispute process with that of an insurance company? Well, I'll tell you one thing, Medicare doesn't spend the 20 million to deny 21 million dollars in claims like I outlined above.
We should make switching insurance companies easier, but that doesn't get us anywhere if they're all embracing the same bad practices. I don't have the option of starting an insurance company out of my garage like you could with a tech startup -- if the incumbents consist of a broken oligarchy, that's what we're stuck with, absent government action. There are so many classic incentive problems in play here, tragedy of the commons, free-rider problem.. it's frankly a wonder that things aren't broken even worse.
With government run healthcare you get the benefit of proper incentives at the cost of government-run bureaucracy, which will always be a little less efficient. That's the tradeoff.. as far as which is more effective, we could look at every other industrialized country which gets better outcomes for a fraction of the cost, or we could appeal to ideology, fears of vague things like "socialism", and absurdities like "keep the government out of my medicare".