There’s a moment in Hell Bent I deeply regret cutting. The Doctor reveals that he’s reassigned the High Council to the sewers, and Ohila remarks that only an aristocrat regards honest work as punishment. That’s the Doctor all over: he knows that the aristocracy must be deposed, but even in bringing it about he reveals that he will always be one of them. If he’s any kind of role model, it’s because he tries to be good, not because he already is. You certainly don’t have to be a white male to play all that – though you can see why it’s a decent fit.
Steven Moffat’s far more interesting quote about diversity in the most recent DWM