A black person living in South Africa (for example) enjoys that same privilege.

That statement alone demonstrates how little you understand the concept. Privilege is a set of advantages one has in society based on their background and where that background places them in society relative to societal norms. Plenty Black people in the U.S. are under-privileged, but a Black person born into a middle-class family that goes to decent schools is not one of those people. Likewise, plenty of White people in the U.S. have privilege, but a lower class White person living in an area that doesn't afford them the same degree of access to standard education, income, and saddles them with the label of "white trash" doesn't. In South Africa, Black people might be in the majority, but they are not running the country.

Privilege is about advantages, not skin color. Learn the difference.