No. You can substitute "Black" for "white", because Black is a specific ethnicity, the one the United States created when it kidnapped millions of people from African and forcibly stripped them of their heritage and culture, and subsequently shared the defining experiences of Black culture, including Jim Crow and forcible segregation, which, along with being crimes against humanity, also helped fuse a distinct cultural identity. Saying "It's OK to be Black" is like saying "It's OK to be Polish" or "It's OK to be Irish", which are things you can absolutely say. Part of the legacy of slavery is that for a large number of Black people, it's difficult to declare an affiliation with their origins in Africa: we took their ethnicity from them in a way nobody took my ancestors Irishness.

It's not a privilege Black people get; it's perfectly linguistically coherent.