As Ms. Reese points out in the first article, there's some canonical material that suggests it's possible. Matt Smith initially thinks he's female and then finds an Adam's apple, there are older references to Time Lords regenerating as female despite having been male, etc. The Washington Post article cites several of them, one of which dates back to 1980 when present-day concerns about diversity, political correctness, etc. were in their infancy. There's apparently a female incarnation of The Master nicknamed "Missy." If the Master, a Time Lord, can change his sex/gender, surely the Doctor could as well.
I can understand the desire of many people to have a female Doctor. If you want to portray more equitable gender relations, having an older male Doctor with a succession of younger and often female companions isn't really helpful. The "Uncomfortable Plot Summaries" website refers to the Doctor as an elderly man who serially kidnaps young women. One of the Christopher Eccleston episodes even hilariously lampshades this by depicting the Doctor interrogated by police after his companion Rose Tyler is reported missing. Ms. Reese's article complains that many female companions are lovestruck damsels-in-distress and not impressive characters willing to call the Doctor out on his crap. Female viewers are likely rather sick of seeing such characters.
However, the presence of Time Lords and Time Ladies implies that the Time Lords are male and female, like humans. The cross-gender regenerations referenced above must be fairly rare occurrences--I've never heard the Time Lords referenced as being by nature hermaphrodites, non-binary genders, etc. that, if these things were fairly common occurrence, would make them. Furthermore, the Helen Mirren comment about a gay black female Doctor comes off as trying too hard.
That said, there is a way to avoid indulging in excessive PC-dom (i.e. the "trying too hard" bit I referenced above) while increasing female representation, especially in more powerful roles. After all, from a perspective of pure self-interest alone, appealing to the growing non-white non-male fandom means more popularity and more cash. My proposal would also shake things up less, avoiding aggravating the old-fashioned fans too.
That way is named "Time Lady Romana." I only watched a little of the pre-Eccleston Dr. Who programs (i.e. I might've seen a couple minutes on TV here and there), but I remember reading about a character named Romana who was a Time Lady, as opposed to a Time Lord. As a Time Lord herself she is a female character who is the Doctor's equal (unlike his companions, or at least many of them) and in the older material, rose to a very powerful position in the government on Gallifrey. Like the Master, she could be an isolated survivor of the destruction of Gallifrey or have something to do with what happened in that episode where a bunch of older Doctors came back and the planet was put in stasis or something rather than getting blown up by the Daleks. Also like the Master (and the Daleks, the Cybermen, etc), she'd be a nod to the older series, much like how old Expanded Universe stuff has been filtering its way into the new Star Wars canon.
Now, far be it from me to let you guys think I'm a white male fan insensitive to the concerns of others. :) On the matter of the role of the Doctor himself, keeping him male doesn't mean he has to be "another white guy." He could easily be played by a Caribbean or South Asian (India, Pakistan, Bangladesh) actor. Here's a bunch of British actors of South Asian background, including Naveen Andrews, who was a major character on Lost and has won or been nominated for many awards. The Afro-Caribbean community in Britain has got Mickey and I think there've been a couple black female companions too; have the South Asians got anybody?
(I'd wished they'd cast Hrithik Roshan as Khan Noonien Singh in Star Trek Into Darkness and he's the physical model for the young Great Khan in the back-story of my novel The Cybele Incident. Could a Bollywood actor be the Doctor?)
Or if that's too radical, another idea is a Middle Eastern type who isn't outright non-white but isn't unarguably Caucasian either. Owing to recent events in Game of Thrones, Alexander Siddig has become available. :) He's originally from Sudan and is of Muslim background even if there's nothing in the Wikipedia article that indicates he's a practicing Muslim. I'm aware of other Middle Eastern actors, but they're all Americans.