First, you would have to specify just what stereotypes of African women you're talking about. I grew up in the 1960s, during Civil Rights and the protests and violence and even have memories of seeing Star Trek during its first run on NBC. Even with that, I'm not clear what stereotypes you're talking about.
(Also, note, Uhura was, at least according to interviews in the 60s and 70s, from the United States of Africa, so she was African, not African American.)
However, if you read The Making of Star Trek by Stephen E. Whitfield and The World of Star Trek by David Gerrold, it helps get a view of what was going on. There was a tremendous effort on the part of all the production staff to make the crew multi-cultural. There was even a big tiff from Pravda that the crew did not include a Russian -- but by the time Pravda was complaining, plans had already been made to include a Russian in the cast.
However, when you look at the stories overall, Star Trek was not nearly as much of an ensemble cast as Star Trek: The Next Generation or any of the shows that came later. Kirk, Spock, and McCoy had a lot of screen time. Other characters were developed as the stories or time fit.
If Asians being fascinated with fencing is a stereotype, I'm not aware of it and such a stereotype would apply as well to the French or Spaniards, simply because in the past, some groups in those countries carried swords or used them in combat. However, it is pointed out that the original intent was for Sulu to have a different hobby every week. First botany, then fencing, then... In fact, in The Trouble with Tribbles, the scenes with Chekov and his interest in the wheat were originally written for Sulu (but Takei was unavailable while filming The Green Berets with John Wayne).
As for Scotty, yes, he's definitely a Scotsman, but that is also part of his character. He's proud of his background and loves his heritage, so he makes a big point of it whenever he can. I'm Scotch, Irish, English, Welsh, and German and, honestly, I've seen this attitude in all those groups and the only group in that list that doesn't take pride in being able to drink all the others under the table would be the English. So his pride is not only healthy, but it's not atypical of a number of groups.
Chekov -- yes, he's proud to be Russian and that was an intentional joke, in part as a salute to the Russians in the space race at the time. Whatever came along, the Russians had it first and best. But can you name an ethnic group or nationality that isn't like that?
Now, back to Uhura. In I, Mudd, the androids tempt Uhura with a young android body that will last forever, so she'll never age. In And the Children Shall Lead, the demon controls her by having her see a mirror showing her as an old woman. Those are definitely playing on stereotypical (at the time) female fears. There's also Lincoln's line, from The Savage Curtain, where he calls her a "charming negress." She, like the other secondary characters, is given the chance to display a few unique traits to develop the character more. In her case, it's her singing - which includes some dancing in Charlie X as she moves around the room when she's singing a song to Charlie. (And one can argue that the singing and dancing are more associated with women than men, or could go with some stereotypes that used to associate those talents with Africans.)
She, and Sulu, and Chekov, and Spock were there to make the same point, which was not just Civil Rights, but equality for everyone, including races we haven't even met yet. They wanted an African on the bridge, as part of the crew. They also wanted a woman in the same situation. (Roddenberry's original idea was to make the 2nd in command a woman, but NBC rejected that.)
She wasn't treated any differently, as a character, than the other secondary characters in the show. She had a few scenes here and there and we saw just how brave she was (in Mirror, Mirror, for example). Just as the other secondary characters in the show, she was given a few traits that made her unique, but not brought forward and fully defined as a character.
Honestly, from what I've seen, unless you saw any of the 60s live and saw Trek as it was before all the movies came into being and read all the articles about Trek and equal rights in the 60s (or, more likely, the 70s, when it was written about much more), it seems like it's hard to see just how ground-breaking it was to have an African woman, a Russian, an Asian, an alien, and sometimes a Scot on the command bridge of an armed vessel like the Enterprise. I know that sounds elitist or snobbish, but any presence by non-white males in such positions was truly a shock to many people.
Since then we've had shows with all-African (or African-American) casts, we've had shows with homosexuals as main characters, and shows with a woman in command of a starship. That these things are accepted so easily now is because of shows like Star Trek (or, for example, I Spy) doing these things when they had not been done before.