"A perhaps more fun question is to imagine, within canon, why the DOCTOR chose to incarnate as a white male, or more broadly, a white male with a British accent."
There is no real sense that the Doctor "incarnates" as a particular race. The implication is that something about their physiology causes them to physically change every few years. They don't seem to get a choice, and are generally surprised, and sometimes disappointed, about who they turns out to be. There has been some equivocation over the years about whether the different Doctors are different beings; or whether they are one person who changes their appearance from time to time. In 325 AD the council of Nicea decided
I would rather frame the question as "How can the Doctor be both a mysterious alien and a British guy who likes cricket and tea?"
And the answer is "They can't be. It's a completely silly and ridiculous idea; as silly as the idea that a box can be bigger on the inside than the the outside; as silly as the fact that the Doctor's life is in mortal danger precisely once every twenty five minutes; as silly, indeed, as the idea that Hamlet speaks his inner most thoughts out loud in Iambic Pentameters." It's a convention of the genre; a thing we accept so stories can happen.
But I can think of other answers, too.
The Doctor had been living on earth for sixty or seventy years when Ian and Barbara first met them. They naturally acquired some of the appearances and attitudes of his adopted planet and retained a lot of those "Edwardian" mannerisms in subsequent regenerations.
OBJECTION: The First Doctor is both an Edwardian gentleman and an alien. Like Susan, they understand some things about Earth and not others. They celebrates Christmas, but don't know what New Years Day is, and have never heard of cricket.
The TARDIS is programmed to blend in with its surroundings. The Police Box form is sometimes said to be a malfunction, but the Doctor has also admitted that they like it that way. Perhaps the Doctor has a built in biological chameleon circuit, and blends in with the people around them. Since they generally go among humans and humanoids, they generally take on their appearances and attitudes. If they spent a long time on Draconia, they would probably become a Draconian, with a liking for Draconian sports and a taste for whatever sweetmeat Draconian kids like to eat.
OBJECTION: We have met many other Time Lords; but never one in an alien form. The first member of the Doctor's race we ever met was a very culturally specific English monk.
We know that Galactus the Devourer isn't really a tall human with purple shorts: he just appears that way to some humans. Skrulls see him as a super-sized Skrull. And I am distinctly enamoured of the theory that when Hobbits meet up with the oldest creature in Middle Earth, they perceive him as an exceptionally jolly and exceptionally silly Hobbit. So clearly the Doctor looks like an English human because we mainly see them through the eyes of English humans. If they had a Thal companion, they would probably look quite different.
OBJECTION: Doesn't work. The Doctor sees themself as having a human shape; the Daleks and Cybermen perceive them as human. Nice idea, though.
We know that there are Greek mathematical formulae on Rassilon's Tomb; that the first Time Lord was Omega, and that the Doctor was known as Theta Sigma in college: which begs the question "Why do Time Lords use the Greek alphabet?". In the days of Usenet a fan named Stephen Moffat suggested that the really interesting question was "Why did the Greeks use the Time Lord alphabet?" He used the same idea when he became a Who writer: it is implied that the Doctor's actual name is Doctor, which is why that word is used throughout the universe to mean a healer, a teacher or a learned person. So, pretty obviously, the Doctor is a posh English human because posh English culture is a kind of Gallfreyan cargo cult. We know that Time Lords drink tea; that they have ranks like Cardinal and Castellan; that they have a collegiate system and even that they regard "24 hours" as a significant time period. So obviously a lot of British and human culture must have been copied from the Time Lords. This is consistent with the idea that "Christmas" is celebrated all round the Universe and isn't simply a Christian tradition. Which explains why the first Doc wished viewers a happy one. Which Moffat was probably quite well aware of.
In Deadly Assassin and Invasion of Time, Time Lords are specifically male (and mostly elderly). They are specifically set in opposition to the all-girl Sisterhood of Kahn. I think it is a pity that this idea got lost, but it did. The first Doctor Who fanzine I ever bought explained that Mondasian prosthetic technology only worked on the male of the species: the Cybermen were really Cybermen. It also said that Susan was a human girl who the Doctor adopted when acting as a personal tutor for an Edwardian family. In those days, fans were allowed to invent canon out of their head.
In all seriousness. I do think that up to now, the Doctor has been primarily presented as a British "boffin". If they became, say, a two-fisted American cop or a patois speaking Jamaican rap artist, that would represent a departure from one of the central tenets of Doctor Who. But I also think that "departing from the central tenet of Doctor Who" is something that Doctor Who does all the time. You might say it is one of the central tenets of Doctor Who.
In 1966 it was seriously proposed that Patrick Troughton should play the second Doctor as a Captain Nemo figure in, er, black-face make up. This would have been a terrible idea for several obvious reasons, but you can see where it came from. "William Hartnell was a Mad Professor in a Time Machine; but we don't want to do that again. What other type of character might be wandering the universe getting into scrapes." Troughton implied that the Second Doctor's characterisation came about as a result of an off-hand remark by Sidney Newman ("play him like Charlie Chaplain if you like!") but in fact the idea of the Doctor as a hobo was quite a canny thinking-through of the basic idea behind the character. Sidney Newman wrote the rule-book; and he didn't think that "The Doctor can only be Caucasian" was an unbreakable Rule of Time.
Twenty years later, Newman proposed that Colin Baker's replacement could be, er, Joanna Lumley -- who at that time was still Joanna Lumley. She'd famously played Purdy in the justly ignored revival of Newman's Avengers, and a not entirely un Doctorish elemental in Sapphire and Steel, so it was not a completely silly idea. So evidently the series creator didn't think "The Doctor can be anyone they like provided they are a boy" was set in stone, either.
I think that some fans objected to a female Doctor for sincere canonical reasons. Not good reasons, in my opinion, but I think there were people who honestly thought "We could have a female Thor and a female Captain America and even a female Prime Minister but everything we know of the Doctor means they have to be a Man."
I haven't heard a single objection to the idea of a black Doctor that does not amount to an objection to the idea of there being a black anything, ever. "The Doctor is British so it follows they can't be black" would be an intelligible position, but even racists aren't generally prepared to be quite so openly racist. Instead, the haters consistently claim that the casting of Ncuti Gatwa has a quality they describe as "wokeness"; that it is part of a process called "box-ticking" and that it is a sign of weakness, childishness and effeminacy on the part of the BBC ("pathetic".) This is precisely the same language that was applied to the casting of Mandip Gill as Yaz, to the casting of Lenny Henry as A Hobbit, and indeed, to the casting of Pappa Essiedu as Hamlet.
But, of course they have a point. (Stay with me.)
The casting of a new Doctor -- indeed the casting of a new anything -- does, whatever RTD says, necessarily have a political dimension. It is not, in fact, possible to "just" cast "the best actor for the job". If RTD says that he was bowled over by Ncuti's reading of the audition monologue then I am quite sure that that is true. But he also knows that if he had cast a white male, the British Association of Racists would have been very pleased, and that by casting a black person, he has irritated them very much. And irritating racists is clearly a very good idea.
And if Doctor Fourteen had been another white bloke, a lot of the usual suspects would be writing cross ranty articles about why it's always old dead white males. It's where we are in history.
If Doctor Who casting has been Colstonised then RTD clearly did the right thing. If everything you do is going to be interpreted as a signal, then you had better make sure that the signals you are sending out are the right ones, not the wrong ones. If the only alternative to virtue-signalling is I'm-a-complete-fascust-bastard signalling, than signal as much virtue as you possibly can.
But I am happy to entertain the thought that some of the people who snarl about box-ticking are struggling to express the thought that they wish the thing was less politically polarised than it has become. I am also happy to entertain the thought that some of the people who snarl about Doctor Woke are trying to put into words an unease about that Chibnall's writing becoming too preachy; that Jodie's Doctor is too inclined to make speeches about things which in the past might have been left to the viewer to infer.
But it should be possible to express such a critical perspective without resorting to alt-right dog whistles.
This week, Tony Blair suggested that the extreme centrist Keir Starmer ought to be less woke, by which he appears to mean