TBF I think the fairytale thing came from a combo of the show's own publicity machine and that Weeping Angels story in Series 5 where the Doctor and River have fairytale related bantz at the end. I've literally found a quote from Moffatt from 2010 saying the was going to give the show more of a storybook quality.
I think while the description is obviously reductive, Moffat's answer is reductive in another, different way - he dismisses it probably because he thinks it takes too simple a view of his work, but the flip side is by dismissing it out of hand, well... it dismisses that... it's true, really.
The fairytale stuff comes from the format of Moffat's series early on: we often see the Doctor through the eyes of the companion, and to Amy he
is a fairytale figure because for years she wasn't even sure if he was real or not, imaginary friend, etc etc. "He was just a story, a game," as Rory said, like most fairytales. This carries through to very, very much of that series and really comes to define Smith's Doctor. What is he in his last story to the people of Trenzalore if not, y'know, a mad fairytale come to life? When you get to the scenes of him doing Punch & Judy and playing with the kids this is really very clear.
Compare this to Rose (this goes for Donna and Martha, too) and what the Doctor is, well, it's pretty matter-of-fact, but the memorable difference, at least among fans, is that RTD's era is a show where the Doctor sits on a sofa and watches Blue Peter during an alien invasion, a show where he gets slapped by the companion's mothers, a show where a pivotal character beat for a major character takes place over a chance encounter as she washes her underwear in a launderette. I think one of the best scenes of that era is just Jackie and Mickey talking absolute crap over chips in a dialogue exchange that's so blisteringly mundane, dull and silly that it feels absolutely real--
JACKIE: And it's gone up market, this place. They're doing little tubs of coleslaw, now. It's not very nice. It tastes a bit sort of clinical.
MICKEY: Have you tried that new pizza place down Minto Road?
JACKIE: What's it selling?
MICKEY: Pizza.
This dialogue is where the kitchen sink drama talk comes from, and people don't talk like this in Moffat's Who. They don't even talk like it in Torchwood or Sarah Jane, or at least not nearly as much - it seems like it was a specific decision by RTD to ground the show for a really broad Saturday night audience, whereas SJA didn't need it because it was for kids and Torchwood didn't need it because it was for sci-fi fans.
Obviously there's crossover between the two - we have The Lodger in Moffat's era and we have the Doctor as a legend woven throughout RTD's era, but I think the comparison is absolutely real. I think Moffat's show ultimately moves away from it - the fairytale thing is specifically a Smith thing and even more specifically mostly an Amy/Rory thing - but it is very much real, and it's a defining pillar of that era. It's reductive because there's a lot else to it, but the 11/Amy and the River stuff is all
absolutely fairytale, and that's fair enough.
The same threads are throughout like Moffat says, but they're handled in different ways- like the Doctor dropping in and out of Amy & Rory's life is a fun game, but with Rose and Martha it's depicted as some awful, life-shattering stuff for the people around them. That's where the kitchen sink stuff comes from, really. We haven't had a character like Jackie in Moffat's era, and she is basically a soap opera character. She could walk onto Albert Square tomorrow and she'd fit right in.