If Gale was batting as he was three to four years ago, that wouldn't be a totally ridiculous shout - you would figure, after all, that he was good for the runs that Ballance/Vince/whoever was offering up. But Windy has not been anywhere near that level in the Championship years. There aren't really any credible external candidates - Morgan is hardly likely to get back into the test side and the assorted Lions captains have not really hit the heights...
I also don't think Root should be given it. We shouldn't overburden someone who is our second best batsman and who plays all three formats. Stokes similar, as well as the temprament issue.
Who does that leave though? For me, Buttler would be a massive call, Broad has been shown to lack the nous required and Bairstow...just let him bat (and keep, if thats what you like). So....if he could hang on for another couple of years, I'd go Jimmy Anderson. We can pack the side with all rounders to keep his work load light, and he does have a decent level of cricket smarts together with the requisite ECB-media profile.