I think you may be confused on the basics of free market system. You are with the absolute majority on that. The goal of the shopkeeper is also to get the very last penny from you for the least amount of product or service. There is no difference between the shopkeeper and the insurance company or for that matter you or every single person posting here. Similarly, the goal of you as the health-care consumer is not any different as the goal of you as a consumer of ice-cream. Your goal is to get the best possible service or product for the least amount of money.
Freedom, the system, is infinitely better than any other; it has always been; it is; and it will always be for as long as this earth and this heaven lasts. This is because it is God's system, and His is unchangeable.
The system is for people to come together freely, to make a transaction, one as a supplier and one as consumer, where each freely bargains for his own interests. In this way they will mutually benefit and they will benefit the rest of the society. The problem arises when one side does not have bargaining power. This problem situation does not arise naturally. Instead, it happens when one side gets someone with guns, in most situations the government, to curtail the freedom to bargain of the other side. In a modern society, this is usually disguised as the exact opposite of what it is. That is, it is disguised as actually something to help the other side, where as in reality is curtailing his ability to bargain for the best possible deal.
The solution is not for socialism, or for a third party, say the government, to try to regulate the one side. Instead it is to find out where and how the freedom to bargain has been removed from one and to restore his freedom.
What is missing and is not understood is that the health-care system in the US is far, far from a free system. It is a social welfare system, that is a welfare system for the special interest. And this is why it is working poorly. Why is that? It is because the health insurance companies, along with every single supplier in the industry, from the pharmaceuticals to nurses to Doctors to hospitals, as well as special interest consumers have got the government to pass laws, enforced at he point of the gun, to curtail the freedom of the consumers to bargain for the best possible deal for themselves to gain for themselves. They have done this by allying themselves with hapless self-proclaimed do-gooders, claiming to be serving "public interests."
For example, you cannot buy insurance from outside your state. I see that you can order books from anywhere in the US. The reason is so that in each state only a few suppliers are left that can easily collude with each other to get higher prices. Or for example, you cannot receive "medical care" from someone who is not board certified. I see that you can have your computer made by anyone, certified or not. Or for example, you cannot buy prescription drugs from outside of the US. I see that you can buy your shirts from anywhere in the world. And on and on and on.