The American version of the British country club began in US by the 1880's. It was a recreational place where exclusive people of similar upbringing, class, and financial resources would meet to network, entertain, and establish business relationships.
However, the American country club lacked the expansive history and myriad of diverse entertainment that the British country club originally possessed, and this is because super huge country estates of the aristocratic British had with them land big enough to play the sports of the rich: Hunting, riding, cricket, and more.
When the American country club developed out of the same interest in providing entertainment to the elite few, Americans began to find specific places where they could enjoy their favorite sports in the privacy of their own class: Everything around them would belong to them, hence, they could pick and choose who goes in and who does not.
It is the elitism that involves it all, basically, what makes the country club play a crucial role in the sporting practices (and other practices) of the rich.