I don't think that way, at all. Capitalism at the core simply means that everyone can own stuff, and everyone gets to keep what they produce. Ownership means that the owner gets to decide how to use the things they own. Capitalism doesn't mean that things aren't shared, it just means that whoever owns the stuff can decide if they want to share. People naturally want to help each other, so they share stuff that they own. People share their things more readily with people they know. Maybe because they might get the favor returned, or they just want to make them happy.

Capitalism doesn't mean competition over co-operation. It simply means that you don't have to co-operate if you don't get equal rewards, or if you don't get rewards proportional to your effort. People hate to co-operate if someone else gets the rewards. People love to co-operate if they get a fair share of the rewards, which is what happens under capitalism.

Owning stuff leads to trading and markets, which leads to the law of supply and demand and the price system. Just by looking at prices of things, people can make rational decisions about what to produce and what to consume. People are pushed to produce things that are scarce and expensive, and they are pushed to consume things that are abundant and cheap. This is a form of self-organized co-operation, where everyone co-operates automatically by just acting rationally. By looking out for themselves, they help allocate resources for the whole community. The lack of this mechanism is one of the major reasons why totalitarian socialism fails. See: https://en.wikipedia.org/wiki/Economic_calculation_problem