Capitalism has own set of problems like unfair distribution of income, unequal opportunities, patent system that prevents spreading innovation. I think it is used only because we don't have a better system at the moment and because people who benefit from it want to keep the status quo.
For example, in a capitalist economy two people that contribute equal amount of time to work can get different rewards for it. And those who get bigger reward would oppose to changes.
Correct, but he will eventually be replaced by somebody who can produce the same output and ask for less. It makes all business sense. Opposing market pressure is unsustainable in the long term.
Then by that same definition, free market is so fragile that it's an impossible pipe dream. Any actor that gains enough power to influence the market destroys the free market by its mere existence.
To take it to its extreme -- you cannot have a truly free market with existence of the State. However, if you don't have the State, there's nobody to enforce your property rights.
But the real world doesn't run on definitions and platonic ideals. There's no perfect "free market" and there will never be, because it's an inherently unstable state in our reality. For the very same reason you can't have world peace by making everyone agree that all conflicts will be resolved by playing chess.
EDIT:
Here[0] you're actually making the same point as I am. Free markets are fragile. They self-destruct if not externally enforced. Unfortunately, the enforcement itself is not immune to market forces either.
Because insurance companies, big pharma and other usual suspects lobbied the government (it wasn't even lobbying; rather, a "mutually beneficial agreement", which is beneficial to everybody except patients) to eliminate competition and lock their profits in various regulated obligatory payments. Being pro-free-markets in not the same as being pro-big-business. Revolving doors are real.
Ironically, free markets are fragile and require government protection to set the playing rules and step away. Usually, the government is not particularly interested in doing that.
(P.S. while the reason for healthcare inefficiencies looks clear enough for me, the case with education is more curious. I think that US education can be a genuine market failure. Compared to healthcare, it is relatively unregulated, and yet people seem to be OK with paying outrageous amounts for something that is not strictly necessary to choose any lucrative career they want. You don't need college education to become a software engineer or quant trader — I've been both without a completed degree, as there are more than enough materials for self-education in these matters these days. There are some exceptions like medicine, but they are that — exceptions).
For example, in a capitalist economy two people that contribute equal amount of time to work can get different rewards for it. And those who get bigger reward would oppose to changes.