Right, I know I'm going to get shouted down, but i don;t care. I wonder if we are getting into a similar situation that required a Brearley recall all those years ago?
Based on some of the decision that Root has made and the impact of captaincy has on his batting that might not be a bad option. Personally, I rather have him scoring runs for the team and not worry about people management side of the house. England need to balance their left/right combination in the team. If the last two tests are going to be on slow/low wickets and Ashwin is fit the number of left hand batsmen could turn into liability, especially if Baristow is not playing due to injury.
If England selector make that decision I would be

My team for the 4th test:
Option 1:
Burns
Mitchell/Robson
Cook (if he is taking paternity leave, pick who is not picked from Mitchell/Robson. Cook needs to be told that he needs to mentor/help the new openers settle down)
Root (will have to turn his arm over for couple of overs of spin)
Morgan (c) - Left field selection similar to Brearley
Pope
Buttler (wk, while Baristow is out with injury)
Stokes
Woakes
Curran/Broad (personally Broad needs to be dropped but I am completely biased against him due to his behavior on the field)
Anderson
Option 2
Burns
Mitchell/Robson
Cook
Root
Clarke/Livingston
Pope (England need to back the young guys in the team)
Stokes
Buttler (c, wk)
Woakes
Curran/Broad
Anderson