Gatiss is somewhat of a logical choice. He has been doing Doctor Who for a long time. He specializes in historical setting though. "An Adventure Through Space and Time" was well regarded over "Day of the Doctor" as the 50th Anniverary Special.
And like Moffat, he's not perfect on his scripts. "Robots of Sherwood Forest" seemed to get mixed reviews depending on how realistic people wanted thing to be. "The Crimsom Horror" also gets a divide. I liked both, but that's because I like that sort of humor in Doctor Who. Other like more seriousness.
I'm so divided on Gatiss. I have a lot of time for him, I love his horror documentaries and he's written some great non/semi Who stuff.
It's just that his actual Who episodes have been a little ropey at times.
Episodes I like: The Unquiet Dead, Cold War, The Crimson Horror (it might even be fairer to say I adore the Crimson Horror to bits)
Episodes that are a bit meh: The Idiot's Lantern, Victory of the Daleks, Night terrors and Robots of Sherwood.
I think he has been handed a duff brief at times, especially with Victory, and to some degree the Robin Hood episode, Moffat sees him as a go to guy for a certain type of episodes, the kind that usually seem to involve a shopping list: "I want Daleks and Spitfires and Churchill and new Daleks and Robin Hood and robots, and flying castles etc etc..."
With this in mind for me at least is the question, if he was showrunner would we get Crimson Horror Gatiss or the Idiot's Lantern Gatiss?
I guess what I'm trying to suggest is that he doesn't seem anywhere near as consistent a writer as either Moffat or RTD, but then again I guess you have to ask who is? Gaiman's got a 50/50 rep at the moment IMO, which is about on parr with Gatiss. Whithouse? I’d say one great story, one good story and one poor story. He has experience of running a show though, but then so does Gatiss.
My choice would still be someone like Joe Aherne, and I do wonder if the BBC are likely to want a succession from Moffat to Gatiss?
I wouldn’t be unhappy with Gatiss. He’s a fan and he knows his stuff and even his poor episodes are never really terrible. I also suspect that, despite his close association with Moffat, that he would produce a very different kind of Who, possibly even something almost gothic in nature given his love of all things Hammer etc. Pertwee is his Doctor which would likely suit Capaldi as well.
It’s all hypothetical though, Gatiss may be top of the list, he may not be even on the list, he may not even be interested.