Saying "black characters are written too broadly in New Who, making them resemble stereotypes" rather ignores the fact that white characters are treated the same way.
Look. This is the problem with trying to raise white people on Sesame Street in order to cure racism: you get a generation of white people who think it's to their credit that they hold everyone to the same standard, and run around operating like the world is one big, happy block party -- people who think they're complementing themselves when they say they're "colorblind."
BLIND is not a moral positive. BLIND is an inability to perceive what the non-blind people around you can clearly fucking see. My grandfather was red/green colorblind. His family also had a strawberry farm. His father used to beat him for not obeying instructions to pick only the RED strawberries and leave the GREEN ones on the bush.
Now, I'm not recommending regular beatings for the colorblind. That wasn't a nice thing to do (my great-grandfather was not a nice person in general, for oh so many reasons). But the thing is, my grandfather's colorblindness? Was a problem, because there is actually such a thing as color when it comes to strawberries, and it's easier to work on a strawberry farm when you can see it.
And there is actually such a thing as race. If you can't see it, you're not doing yourself or anyone else any favors. There are cases where you can give the EXACT SAME script/character arc/iconography/etc. to a white performer and to a performer of color, and the overall effect WILL BE DIFFERENT. Race is real. People respond to it, often on levels they aren't entirely aware of. So it actually misses the whole entire point of discussing race and racism if your sole defense is "but we're just treating them the exact same way we treat white characters!" It may be true, or it may not be true, but either way it's singularly useless.
Some fans seem to find gender easier to understand than race, so think of it this way: if there's a character that isn't very bright but always uses sexuality to manipulate other people, does it make a difference if that character is a man or a woman? Isn't it more of a stereotype in one case than in the other? And if some writer or producer said, "Oh, it's not sexist -- this is just what we were going to do, and we thought we might hire a male actor, but we went with a woman instead, so we kept the same stuff!" that doesn't magically make her not a sexist cliche, does it? If they'd cast a man, the character would read one way; when they do cast a woman, it reads differently. Same character. Different, because of the baggage we bring surrounding gender. If you were somehow magically oblivious to any and all gender issues, you might not notice that. But you wouldn't thereby be a better person than the rest of us. You'd just be oblivious.
Unfortunately, in our culture, we are conditioned to see white people as Real People, and black people as sort of thin slices of people, operating in one of a very few available modes and with only a very few emotions and interests. Therefore it's just different to write a white character "broadly" versus a black character. It's not enough to write the black character "just like" all your white characters, because race is not invisible to most of us and it doesn't have no consequences. In order to challenge people's already racist assumptions about black characters, writers have to work that much harder, and they have to work not blind. They have to work with their eyes open and their brains engaged and with the awareness of subtle signals and context and connotation that anyone who writes for a living should damn well be conversant with. To do less than that is to write lazily, to write foolishly, to write contemptuously of one's characters and one's craft, and to do all that because you can't or won't go the extra mile to bring race into the universe of stuff that factors into your writing does, in fact, have racist implications.
"Colorblindness" may be one's reason for making all of those mistakes, but it isn't an excuse, and it doesn't magically make the product impervious from criticism. Be less blind.