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.