I'm always in favour of historical time periods being represented as accurately as possible. Moffat is okay with historical fantasy, as evidenced by the Viking episode last season. It's one place where my sensibility doesn't overlap with his.
The notion of 'improving' history by ethnically diversifying it is a bit of a silly statement. The vast majority of people simply didn't move around the globe until the 20th century. It's not to history's shame that cultures were not, for the most part, ethnically diverse, it's just a fact of how technology progressed. It is to our shame that typically different ethnicities often had different social standings where they did co-exist but that is something that should be portrayed, not shied away from. One could argue that the ethnic diversification of the Americas had catastrophic consequences for the Native American races and cultures that already inhabited that continent before the migration of Europeans. Would history 'be improved' by portraying an Aztec culture of black and caucasion people?
ON THAT NOTE, as Mullet touched upon, if we look at the 1960s Doctor Who serial, 'The Aztecs', we did see an Aztec culture made up entirely of white actors. Historically, casts were not diverse even when they should have been. I daresay they still aren't in many cases. As Mullet said, the casting of ethnically diverse actors is a tactic of positive discrimination to try and counter-balance this historical trend.
Also, how many future or alien societies (including the Time Lords) have been portrayed as entirely or nearly entirely white? That is nonsensical. Making such casts diverse isn't 'political correctness' (people who use that phrase typically don't even know what it means) it's logical. In most areas racial diversity in the cast is essential for a credible portrayal, if for no other social reasons. Indeed, if the human disapora continues at its present rate our future societies should be totally diverse.
But I really dislike the notion of 'judging' history full stop, let alone judging it against modern standards. Again, the idea that history is 'wrong' and contemporary society is 'right' is myopic and ego-centric in the extreme. I thought that part of Moffat's speech was childish and ill-considered. I'm sure he was speaking off the cuff of course and with a bit of analysis would agree. Of course, some aspects of history, such as slavery (which in fact still exists in the world), militate absolutely against our contemporary values but preaching against history has no worth as a pursuit; studying it, understanding it and learning from it does. Presenting a fantasy version of it is, to me, the worst possible option because if we censor history, or dress it up to make it appear more palatable then we cannot learn from it. There's no better critique of racism than an examination of the historical slave culture of North America.
It's a more complex issue than Moffat simplifies it to in those soundbites but it's also a much more complex issue than some of the idiotic comments made by Moleskin in this thread. Ethnic diversity in casting is absolutely a legitimate issue and something to be considered. But I think an accurate, and, if necessary, 'warts and all' portrayal of history is also important and beneficial to our culture. Sugar coating, or modernizing history is to negate one of the biggest tools for learning and improvement we have: the past.