My thought on why mandatory health insurance wouldn't work originated with an investing book, Unconventional Success by Swenson. (I learned a lot of new ways of thinking about the world once I learned about investing.) In it Swensen argues that stocks are good investments because management's interests are mostly aligned with shareholder interests. Government bonds are OK investments, because the government wants to promote stability and prosperity, and yet it doesn't really want to pay you back, so it is neither for nor against you. Corporate bonds are bad because management's interests are opposite of yours.
I applied the same method to health care. First, I realized that universal health insurance is like corporate bonds: bad for you. The insurers make money by taking as much as possible from you, and paying as little as possible to health care providers. At every point it is in their interest to delay, obfuscate, negotiate, and make lower money they owe while continuing to demand ever greater payments up front. Opportunity for shady deals with providers, manufacturers, and suppliers abound so you will never be able to get a free market effect by competition through insurers. There is no alignment of interests between mandatory insurers and you, what so ever. The very best theoretical outcome is that you get the same care for the same price plus the labor, materials, and profit of of the insurers. In other words, the same care for a higher price is the best outcome as a system (Obamacare did extend healthcare to people who couldn't have it otherwise, which was good, but it made the system overall worse).
Next, obviously you are interested in your own health. This is the free market we all know and love. But I was surprised to realize that government is, if any thing, at least as interested in your health as you are. Not always or in exactly in the same way as you personally, but in aggregate it is in the interest of the government to keep its citizens mentally and physically fit so that they can maximize GDP and be good little soldiers if needed. That war on drugs, mandatory vaccines, seat belts, the cigarette and sugar and booze taxes? Not actually judgementalism for its own sake. The forces of big government are concerned that many citizens are losing their health edge. At that point I realized that the government would probably be more incentivized to provide effective and efficient health care as measured by aggregate data than the private free market would. You might argue against government health care simply because you are against big government, but if you say you oppose it because it would be less efficient than the individual free market you will be wrong. The government has strong incentives to provide the best aggregate care for the lowest cost in a timely manner, quite likely stronger than your own personal incentives.
Now to the problem that the government's interests are mostly but not entirely aligned with your own. The government only cares if you function well. You want to enjoy life, which in health care terms is maybe about 80% the same. We still need to give individuals the chance to get more, less, none, specialized, or different healthcare whenever they want and are able without barriers beyond availability and their own ability to pay. So we need a way to work that in. The problem with unconstrained government health care isn't inefficiency, it is overefficiency.
I worked for a few weeks with a poor Canadian guy who had a hernia, but his system couldn't schedule him for surgery for months for any reason. He was walking around in pain and on meds. To me that is a little bit of a failing. It's nice that he got low cost surgery, but the long delay leaves a little to be desired in my book.
My favorite (from my very shallow perspective) health care system is China (I have heard South Korea is similar but never been there). Obviously it is highly subsidized. It is astoundingly fast, you get same day service for virtually anything, and if you show up with an ear infection and no appointment you will see a PA/NP equivalent and walk out with a prescription within 30 minutes for the entire process. To my mind, beyond heavy subsidies, the defining characteristic of the Chinese system is that you must pay up front in cash before you receive any health care at all. Sure the cost is very small, but you don't get in if you aren't prepared to pay it. Pay $3 to see the doctor-type-person. Need lab tests? Take $3 to the payment window then get them done. Then return to the doctor, who gives a prescription. Back to the payment window with $2 and then go get the prescription filled. To my mind the cash-only pay-in-advance-as-you-go system keeps prices transparent, patients attentive, and generally acts like grease on an axle. I see it as beneficial. There are some problems though, for example apparently the providers get paid based on the cost of what they provide. This has made doctors almost always recommend C-sections over natural births because they get paid more for them. It is not a system I would want in its entirety, but there are many parts I like.
Then there is what the USA has. Slow like Canada. Perverse incentives like China. No price transparency to consumers. No price negotiation by government. It seems to have copied the worst elements from all of the world's other systems into a giant mess. If I had $10,000 to spend, I bet the quality(times)quantity of care it would buy in the US would be the actual worst in the world. To fix that I propose that a system that is roughly 80% government, 20% private/individual will be best. I have no idea of details needed to implement that, but that's where I would like to go.