The funny thing about this scrap of criticism is that Pratchett probably can leave his brain on a hook at the door. Yet he's the one complaining about having to do it. Yes, Doctor Who is pure entertainment. It is not a story of "functional magic" as TV Tropes might call it, but rather a series of excuses for the things that occur. However, these excuses are completely justified within the Doctor Who universe. The show is just a rare example of a fictional world where the protagonist is, quite literally, capable of anything.
Now, for Pratchett this might come across as a bit of a case of "Boring Invincible Hero", to evoke another trope, but the Doctor's limits are there in the show, they're just not in the technology, but rather in his ideas. Yes, the Doctor is a genius. He's lived for more than nine hundred years and still has a youthful mind and a spring in his step. The drawback of this is his insanity, which seems to be, ironically enough, the thing that Pratchett's missed in his comments or his frustration.
The Doctor's flaw is in his ideas. The Doctor Who series has always cast humans as a kind of "creative" species, made for dealing very well with emergency situations. This is why the Doctor needs a human companion, and on the occasion where he has none he often ends up in hot water - they fill in the massive hole in his problem-solving. That is, common sense, which is something most mad geniuses don't have.
Every episode of the modern series, in one way or another, exploits this limitation of ideas to propose a challenge to the Doctor and create much of the tension. Yes, he usually wins, but not without help. And it is that limit that you have to look for to see the show's real depth, which is a bit more subtle than the Doctor's antics and all the superficial technological wonders that are just there to be cool.
Doctor Who takes a leaf from Shakespeare's book in the sense that its themes are more universal than specific to the setting, and it is that which makes it different from most sci-fi. The best of sci-fi often investigates "special themes" of the future, which humans do not have to deal with in the current era. Doctor Who shares more in common with "low media" in that it is about looking at human pain, the struggle, and whether the Doctor likes this or not, the concepts of "good" and "evil" (something we've seen in films like Lord of the Rings). But not all sci-fi has to be high culture, and not all of it has to look into strange and alienating themes.
Doctor Who is a show about humanity, even if the central character is an alien, and in this way it has great value. It should not be discounted as a science fiction story simply because it has more common, modern themes than futuristic ones - the science fiction is there, but simply used as a backdrop and framework for the classical challenge. The Doctor uses science to solve problems, not magic, so we cannot class it as fantasy. That would simply please Pratchett's ego rather than solve the debate or be fair to the show, which remains a brilliant example of good television.