A Black Yeoman would fit into the racial and gender stereotypes of the era to keep TV execs happy. The character of Yeoman Rand or her replacement would basically be a secretary for Captain Kirk. The character only appears on the bridge to hand Kirk a cup of coffee or hand him some report.
I am wondering how Roddenberry would write the character of the White Conmunications Officer. Perhaps his wife Majel Barrett gets the role instead of playing ship’s nurse?
Another idea I have is make Uhura Hispanic to keep the multi ethnic cast. Uhura could become Lt. Garcia for example. You could still cast a white actress but it could be implied that the character’s family originated in South America. A better story could be that the Uhura character is Cuban to play along with Chekov being Russian ( in the future our enemies are now allies).
I think that in the pilot they had a navigator who had a Brazilian mother and American father, Jose Tyler, so that would definitely work.
Roddenberry wanted to get Majel Barrett in any role no matter how small so she could have played absolutely anyone.
If they did have a black actress as a Yeoman and had her be the same character throughout the series then there would be a possibility for character development along the way.
Perhaps if there was a white communications officer then we could have a black actrees play either the Nurse or perhaps instead of the Sulu character and then have an Asian Nurse or Communications officer rather than have an entirely white cast, much like in The Cage.