If Doctor Who were really interested in counteracting ethnocentric thinking, and in getting audiences to really think about racial relations in the 21st century,
Well, that's the essence of the criticism that I and others are bringing to the table: That Doctor Who has diversity as a virtue amongst its themes, yet often does not live up to these themes.
the writers would bring the old "historicals" back to the show. It's a shame, I've always thought, that the show became so monster-obsessed, when early episodes like Marco Polo and The Aztecs were fascinating.
In fact, first have the next incarnation of the Doctor be a non-white character. Then, have a larger proportion of the episodes take place in various periods and countries in our world's history. Then, we can see how those places and times would react to a non-white character bossing people around and "walking around like he owns the place" (something the privileged and white David Tennant incarnation suggested has always worked for him.) This could be a fruitful avenue of storytelling, a method of commenting on our present by exploring our past, something the usually light-and-frothy Doctor Who has generally shied away from doing.
I completely agree with you that featuring a non-white incarnation of the Doctor, and how he would cope social reactions to his perceived "race," would be a fascinating avenue of storytelling.
By the same token though -- I don't think Doctor Who (in its modern incarnation) is meant to be a show about race relations, and I worry that exploring this topic in too much earnestness would undermine the fundamental dramatic conceits of the series. At the end of the day, modern Doctor Who is an action/adventure superhero story about an alien who does not interfere with conventional history but intervenes when the fantastical comes into play. It is a series that is, first and foremost, designed to appeal to eight-year-olds (and the inner eight-year-olds of adults) who want adventure and escapism. Doctor Who is meant to be a mostly light-hearted show that occasionally has bits of earnest, darker drama to mix things up; most episodes are meant to have tones like "The Lodger," with the occasional "Vincent and the Doctor" mixed in.
So I would argue that the challenge would be finding a way to approach the topic of race relations and a positive depiction of diversity in a manner that is honest, that does not reflect white privilege (as you correctly noted the Tenth Doctor's advice to Martha in "The Shakespeare Code" reflected white privilege), yet also does not detract overly-much from the primary goal of featuring mostly light-hearted fantastical superhero stories for children.