Having read a few interviews with Gaiman recently where the subject of his Who episode came up he made it clear that he found writing just one episode a huge endeavour, it really is a lot of work, especially if you're writing novels/comics/other film and TV at the same time. Frankly I'm not sure Gaiman would/could commit to that--plus to add that being a writer doesn't automatically make you a good show runner.
Oh and as a final point re Gaiman, the other problem is that his Who era would probably be too similar to Moffat's anyway in terms of its fairytale nature. Whether you like Moffat Who or not (and I do) the next showrunner's era shouldn't be more of the same.
Personally I want Moffat to do one more series at least, but maybe not more than that. Much as I felt with RTD I'd quite like it if he still wrote for the show in future but much like RTD I doubt he will
I think it would be very interesting to see Moffat chart the initial course of Doctor 12, but then someone else take over, we've yet to see--in the modern era-- the same Doctor through two different show runners' eyes.
As for who should take over eventually, who knows (pardon the pun) it's not like we have any say in it, and it might be a name we recognise it might not. Every name in the hat has pluses and minuses;
Toby Whithouse, I never saw Being Human but his Who work's been pretty good, well until A Town Called MErcy anyway, the only dud note of 7A for me.
Gatiss , knows Who inside out and has a lot of experience, though oddly before this year I’d always enjoyed his non Who writing more than his Who episodes, but then in 7B he gave us The Crimson Horror and Cold War, the two best episodes aside from The Name of the Doctor, so he’s higher up my list than he would have been six months ago! Given his love of the horrific we might get a more gothic Who from him.
Chibnall, there was a time I’d have potentially stopped watching if he took over. His Who episodes had been a bit bland and as for early Torchwood…but I really liked the work he did on 7A, and you can’t deny he’s racked up a lot of show runner experience. Still wouldn’t be keen but as with Gatiss I’d be far less apoplectic than I would have been at one time.
Personally I’d give the job to Joe Ahearne, oodles of experience as a director, writer and executive producer plus he hasn’t worked on Who since 2005 so it would be less of a case of the BBC promoting from within Who.