Edit: Capitalism is a far more complex system than simple private ownership and trade. A critical feature is artificially reduced liability. Which goes beyond simple freedom and gets into practical considerations.

The fact that capital ownership in a cooperation does not expose an individual to legal liability for every action of that corporation is what let’s you have capital markets. A banker may loan a car dealership money with minimal concern beyond the risk of that money.

Without that protection the engine of capitalism breaks down even with private property rights. But, even simply applying strict liability to management would cause significant breakdowns. Picture the CEO of Five Guys going to prison if even a single person got food poisoning via negligence. Are they going to try to expand past a single burger joint?

PS: In practice this seems to work well. But, the ability to have legally enforceable contracts etc goes beyond simple trade. After all people would buy and sell used goods in socialist countries, but such trade did not make them capitalist countries.