I may be sorry that I'm wading into this, but in many cases, I think an ALL-black cast would be seen today as a contrivance, especially for particular stories. Even true race-blind casting of the leads in certain shows could change the story. For example, My Fair Lady is very much about rigid class structure and bigotry, but it's not about race. Higgins has to be a white CHARACTER, even if played by a black actor, because he was a member of the privileged class in what was then an almost all white society. Eliza has to be a white CHARACTER because a black Eliza could have spoken the Queen's English and still could not have moved up because of bigotry. Neither could her father. OTOH, a rewrite of the story to move it to NYC and move the date up to at least the mid-20th century could work with Higgins and Eliza as black characters. Perhaps Higgins could be testing a theory that high-class speech can overcome racial barriers. However, "The Rain in Spain" wouldn't work any more.
Many stories can work well with black characters originally written as white (e.g., Willy Loman, Spiderman, Mary Poppins), and there's nothing wrong with doing a rewrite for other shows to feature music that might appeal to young black theatergoers (e.g., turning The Wizard of Oz into The Wiz). However, I think that, in the 21st century, casting a whole show with black actors for no articulable reason risks offending people in more than one cultural community.
Audrey, the Phantom Phanatic, who nonetheless would rather be Jean Valjean, who knew how to make lemonade out of lemons.