But they’re not ‘making period figures black’, they’re just casting colour blind. No one at the RSC is claiming Henry V was black. As Shakespeare only wrote about two non-white roles, a major National institution whose work is overwhelmingly based on his plays either chooses colourblind casting or opts to be whites-only as regards actors.
Some people really dislike colour blind casting, or they dislike it in more historical type settings. Look at someone like Samuel Jackson complaining that the wrong type of black person was being cast in certain roles - not American.
I find that people often see television in general quite differently from the theater, unless it's a very non-realistic setting. That's not an unfair thing, television and film often go to a lot of trouble to immerse the viewer in a setting that feels realistic. Unlike theater where there is less realism and also a long tradition, especially in small theaters, of roles being played by whomever was avialable.
But it's also not like people are consistent. There was a big kerfuffle in the city I lived in when a small theater production of The Mikado, using very traditional costuming and setting, had European actors in the roles. Keep in mind that I live in a place with a very small Asian population, and an even smaller Japanese population. So no race blindness there even though it would be totally historically accurate for the play and also just realistic in terms of a small city theater production.
People feel like, it's ok to complain about and ultimately shut down a small production of TM because of ethnic representation, but Anne Boleyn should be played by anyone, (maybe even a man!) It doesn't create goodwill.