Private clubs engage in commerce.
The reality of the matter is that most people like being around their 'likes.' A large portion of blacks associate with blacks, whites associate with whites, Germans with Germans, Spanish with Spanish, Korean with Korean, etc, etc, etc.
It is only through trade, commerce, and the markets that individuals interact with 'unlikes' on a consistent basis. Free markets, regardless of what any stupid fucking law says, integrate society and bring people closer together. It is precisely the absolute voluntariness of human association and separation - the absence of any form of forced integration - which makes peaceful relationships - free trade- between ethnically, religiously, linguistically, racially, or culturally distinct people possible.