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.

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.