If the cost is 100 for 10 cases, then it's going to be 10 for 1 case. Cost per person is still the same. You need fewer doctors, nurses, and medical devices to treat fewer people.
Ok, I see where the problem is. The 10 people requiring the 100 treatment in your example will cost 1000. That's one dollar per person, not one tenth. For the group of 100, 1 person will require the 100 treatment and it is still 1 dollar per person. You very obviously made a mistake and I incorrectly assumed it was in your calculation of the costs for the second group. The underlying point is still the same though, and you should be concerned that such a simple relationship is not intuitive to you. I'm not going to apologize because I still feel that this is indicative of emotional reasoning on your part. If this sounds condescending then that's only because this is about as simple as math can possibly get and yet you still failed to recognize the glaring logical contradiction.
It does make sense. A smaller group will have a correspondingly smaller number of people who will end up needing treatment, so the expected value for an individual's cost doesn't actually depend on the number of people in the group, just the probability that any specific person will end up requiring treatment and the cost of the treatment. Maybe you are confused because you chose a specific example wherein the expected number of treatments required is less than one? Are you thinking that because it's not possible to have only a fraction of a person require treatment that we need to round up to 1 person? That's not how probability works and if you simply extend that logic to a group of a single person then your error should become immediately apparent. While it's true that only whole numbers of people can get sick, when you take repeated samples of small groups you will find that occasionally one group member will require treatment, but much more commonly than that, zero group members will require treatment. When you weight each case by its relative probability of occurring, then you get the same expected cost per person that you calculated for the large group.
Your lack of intuition about such a simple linear relationship should be extremely concerning to you. It suggests that you are approaching the question with an emotional attachment to a particular outcome that is blinding you to the extremely simple and intuitive correct answer.
There's a lot more to unpack here, but I don't have time to address it point by point. You are making a lot of assertions that seem much more reflective of left-wing ideology than actual conditions on the ground. You suggest that consumers demanding better is a solution to public systems that don't actually meet consumers needs, but the very nature of public goods ensures that no individual consumer will ever be in a position to make a demand that administrators will have to prioritize over competing demands from producers and public employees. The nature of democracy gives outsized influence public sector unions and interest groups that can then turn around and contribute a share of those benefits back to politicians campaigns. In contrast, a competitive market affords consumers to actually make demands with some weight behind them. A producer has to take a consumers demands into account lest that consumer simply take his business to a competing producer. The weight of the empirical evidence affirming this fundamental difference between public and private systems is completely overwhelming, and your resistance to the idea that capitalism is a far better system than socialism for the vast majority of goods suggests an emotional attachment to an ideology that has very intentionally been sold to the public in such a way as to make uncritical advocacy for that ideology very self-flattering to those very advocates.
I know that I didn't really address the middle part of your argument but let me assure you that those ideas are just as contrary to reality as any of the ideas that I chose to address. I might come back to those in a future post if I find myself with both the time and the inclination to do so.
A risk pool being split among multiple companies doesn't actually increase the average amount that each group would have to pay. For the group of 100 in your example, only one tenth of a person is going to have to get one tenth of the expensive treatment, so each one still has to pay 0.1. As long as each underwriter has a sufficiently large group to ensure that the average cost incurred by its members is sufficiently close to the population average, the relative sizes of those groups isn't going to affect their members' premiums. Insurance administration is not an economy of scale so competition in the marketplace would ensure downward pressure on prices. In contrast, a state provider would have little incentive to cost-cut or innovate since they wouldn't have to earn people's business, and any public option would necessarily be attempting to balance the interests of multiple groups (such as public sector unions/government employees, health care providers, and health care consumers) at the expense of the consumers whose interest would have been prioritized in a competitive market. Taking that into consideration, it's doubtful that the lack of shareholders in a public system would represent enough of a benefit to actually lower costs for consumers. Americans have enough experience with poorly managed government programs to intuitively understand that a socialized health care system isn't going to be an improvement over our current system.
And yes, since national defense and law enforcement are desirable services that can't be provided by a competitive market, we are fine with the government providing those specific services in a constitutionally-limited way. That preference is actually more ideologically consistent than your preference to socialize our healthcare system unless you're going to advocate for abandoning capitalism entirely.
D and E are simply the result of healthcare consumers choosing to spend their money on care that wouldn't even be available under a public system. Taking that into consideration, "look how much of our GDP we spend on healthcare" is not the glaring indicator of a broken system that people make it out to be. People are allowed to spend their own money.
How is that policing the police? Are disparities supposed to be evidence of something nefarious going on? Given that there are fundamental distinctions between members in different groups (otherwise they would be in the same group) and almost certainly many other non-fundamental distinctions that correlate with the group-defining distinction, is it not entirely plausible that there should be disparities in police statistics even when police act appropriately 100% of the time?