Terry, I would counter, what makes government employees so special?
You are not eliminating the middle-man. You're trading one for another. Just because Medicare, Medicaid, the VA, etc. are govt.-run single-payer systems hasn't made their formularies any simpler, broader, or easier to challenge.
In fact, many people would argue, worse.
The answer is, as it almost always is universally, get the government out of the way of any decision-making. Trust The People to make their own decisions. Will some people make bad ones? Yes. But overall, it is so far the best approach we've found organizing society to maximize the most people making the best decisions for themselves possible.
The way to do that is reduce govt to simply a subsidy mechanism, and let the rest of the "system" work through the private market. Just as education should be handled through funding vouchers, so should health insurance.
Tax unequally to raise the funding, according to politics, distribute the funds equally for education and health insurance. Let parents decide where to educate their children, individuals what health insurance best meets their needs.