Alright, I'll presume good faith since you've responded (this will take a couple posts), but capitalism is not merely a system where private individuals own the means of production. It is a specific social relation between wage-labor and capital, under conditions of systemic market dependence. Defining capitalism purely in terms of who owns the means of production neglects a vast body of scholarship working with a far more robust and systemic description of capital's "laws of motion," and reduces it to a quibble over property rights. Even if we replaced individual private ownership at scale with cooperatives, systemic market dependence remains, i.e. I still have to get a job and secure a paycheck or starve.