I just finished season 2, and I'm pretty underwhelmed. So far I feel like I'm watching a bad show with some good moments. My friends keep pushing me to watch more....I might write an LTTP once I catch up.
There's a definite and huge uptick in budget and quality from where you are now. I'd say you're over the hump, the bumpy bit of the ride. Stick to it, let us know how you feel about it!
Are most of those even Moffat directed episodes?
Kids are, generally speaking, a strong running theme that he likes to use, yeah. All of the RTD-era episodes I evoked are written by Moffat, and his fingerprints are on all episodes of everything from Series 5 on.
Obviously The Empty Child/The Doctor Dances is all about a teenage mother and her young son, and features a ragtag gang of kids.
The Girl in the Fireplace focuses on the Doctor meeting a girl as a child, then again later in her life, the childhood encounter having a profound effect on her.
Blink actually doesn't feature any kids, but it is worth noting that Moffat adapted it from a short story he wrote called "What I did in my Summer Holidays, by Sally Sparrow." That was a ninth Doctor story, is basically structurally the same as Blink at its base level. In it, Sally Sparrow is 12 years old and, again, gets a glimpse at her own post-Doctor future, significantly changed by her encounter with him as a young girl.
Silence in the Library/Forest of the Dead features CAL, Charlotte, the little girl trapped 'inside the TV' with a slightly hapless dad and the creepy Doctor Moon. Again, the Doctor appears to her through her TV (as in Blink) and interacts with her.
Amy's story is ultimately a sort of culmination of the stories expressed in the previous three Moffat episodes, but explored across multiple series and episodes with young Amelia.
There's obviously lots of stuff with young kids outside of Amelia, too, like the kids in The Beast Below and such.
Moffat's original intention for Series 7 was for Clara to not die in the Christmas episode. It would've been that Clara who travelled with the Doctor, the second one we meet. The production team got cold feet about a non-contemporary companion at some point and then adjusted Christmas so she'd die, and he then meets her in the present day.
The original plan was for the brother & sister duo from Victorian London to come on travels with the Doctor and Clara and feature in 3 or 4 episodes in Series 7b. If you wonder why they return to Victorian London so fast after the Christmas episode with The Crimson Horror, it's because when S7 was mapped out, The Crimson Horror would've been Clara's traditional 'return home' episode, where she checks in on her real life. The series was at a point where planning had got to a point where they couldn't change the episode when they made the modern Clara switch, so they kept it. One of the losses from that episode was presumably the kids, which is why the cliffhanger leading into the Cyberman episode is so tacked-on.
They probably would've yanked the kids from the Cyberman episode too, but Gaiman is an incredibly busy man and likely didn't have time for the massive rewrites, so they worked it in. The original plan for Series 7, though, Moffat has said in DWM, was for the kids to have a much larger role.
On that topic: It doesn't really matter too much what Moffat wrote, he oversees the entire series. That's his job. For instance, Gaiman didn't choose to use the kids, Moffat told him "Clara, the two kids she's a nanny for, Cybermen, an in-universe explanation for a new Cyberman design." Gaiman then worked with that. When Gaiman returned his script, Moffat read it and wanted, for instance, more abilities added to the Cybermen. So the bit where they remove their heads and detach their hands was Moffat's, not Gaiman's, for instance. This is standard practice.
Aaanyway, I don't really mind the kids as a one-off thing, I'm just not really a fan of kids travelling in the TARDIS, as I often think giving child characters something to do or putting them in jeopardy is just a waste of script space in many cases when the episode could function perfectly well without them.