Someone with $10 million inherited capital is more likely to succeed at marketing their product — regardless of the quality or positive social impact of that product — than someone born with nothing.
I would define "capitalism," as opposed to "free-market economics," as a system under which the person with $10,000,000 funds the business of the person with no money. The entrepreneur specializes in satisfying customers, and the capitalist specializes in taking the risk of failure. It's actually a surprislingly mobile system.
Socialism is a system under which government experts, rather than entrepreneurs funded by capitalists, decide what will be produced. Notions of what is "fair" are not "socialist," nor are they inconsistent with capitalism. Indeed, one can make the argument that providing subsistence incomes for everyone, who then buy what they want from entrepreneurs who respond to their demand, is the best thing that could happen to capitalists.
The Scandinavian governments go to great lengths to explain that they are capitalist states. The safety net is extensive, but it is not "socialist," because the entrepreneurial decisions are made by entrepreneurs funded by capitalists.
The "market" certainly is the central point of disagreement between socialism and capitalism. But the disagreement is not over who gets what; it is over who decides what is made. Capitalists do not, qua capitalists, believe that the market should decide who gets what and how much. If the Government gives out food stamps, capitalists will fund farms and grocery stores. Some will make a lot of money doing so, and they won't complain at all that the food stamp program changes the system from "capitalist" to "socialist," because the capitalists and entrepreneurs are responding to the the modified market.
There is no doubt that "laissez-faire" market economics can have disastrous externalities. But wise capitalists understand that the government can make bad practices unprofitable by changing the "market." Banning a product or a production method to protect the environment is not "socialism." It's just customers, as voters, announcing that they will not buy from producers who do things that the customers, as voters, have agreed should not be done. The rule takes no skin off the capitalist's nose, because the capitalist only needs fair competition under uniform rules. If the government adjusts the market to make capitalism less damaging, capitalists have no gripe. Some capitalists will gripe, but not because they are capitalists. There are short-sighted people in every walk of life. Just because one is a capitalist does not mean that one knows what "capitalism" entails.