Why, then, is the Doctor so apparently suited to Christmas, even though the connection was only made once in the show's first 26 years?
Maybe it's because Christmas is a time for "Peace on Earth" - except that the Doctor brings precious little of that! He may have made powerful statements against war (eg in The Zygon Inversion), but it's hard to equate John Lennon's "War is over if you want it" or Jesus' various teachings with Xmas quotes like "no second chances, I'm that kind of person".
No, the Doctor's not Jesus, Unless you're watching Last of the Time Lords, which wasn't even a Christmas special. Is he, then, Santa? It's certainly been suggested over the years – the red bicycle comment – but really, they're not that similar. The Doctor doesn't turn up, leave presents, eat food and then go. It's just not even remotely like his MO.
I think the key might be in comments made by various Doctors over the years but particularly the third and fourth. There is no point in being grown up, we are told, unless you can be childish sometimes. The Doctor is serious about what he does, but not necessarily about the way in which he does it.
This is the time of year when we decorate our houses with tinsel and flashy lights and baubles. We encourage our children to sing songs about reindeers with red noses and tell them to anticipate a fat man climbing down the chimney with presents. It is, surely, the most childish, or maybe child-like, time of year. As Andy Williams nearly sang. The time of Band Aid, sure, but also Lily the Pink. Carols from King's, but also Mr Blobby.
The doctor may spend Christmas sending off the likes of the Sycorax and the Master and the Great Intelligence, but he also does it in a fun way. These 10 episodes have seen him battle Christmas trees and flying sharks; team up with Kylie Minogue and Nick Frost; despatch baddies with a satsuma and gain acces to their lair by impersonating Sherlock Holmes. Where else would we find Jessica Martin voicing the Queen waving at a flying Titanic or June Whitfield pinching the Doctor's bum?
Russell T Davies was adamant that Who's Christmas Specials should be Christmassy, so served up deadly Santas, trees, baubles, stars abd angels before visiting the heartland of Victorian England. Remove the setting, though, and The Next Doctor is light on Christmas stuff. The End of Time really has to crowbar it in - and arguably not very well. A new approach was needed, but it was perhaps surprising that Steven Moffat chose to amp up the Christmas content by about 400%. Homicidal snowman? Check. More Victoriana? Check. People singing Christmas carols and dancing to Christmas number ones? Check.
For his first one, Moffat outrageously stole not only the plot but the title of Charles Dickens's A Christmas Carol. Could he get any Christmassier? The Doctor, the Widow and the Wardrobe may not present a traditional Christmas story, but it might be the most Christmassy Christmas special of all. What about that extended sequence when Matt Smith shows the children around the house he's created? Then they go to a world where Christmas trees grow complete with baubles. This all led up to the 2013 Christmas special, in which he consigns the Doctor's final official incarnation to his grave… In a town called Christmas. Then, the next year – Santa.
The children's own programme which adults adore has implicitly and explicitly sold itself as an advocate for childishness and childlikeness since Troughton first clowned around. No wonder it suits this time of year so well.