Yes, probably the team won't peak with Cook as captain. the very things that make him a superb opening bat (conservatism/denial) are the things that he needs less of as captain. His management of the spinners is a key pointer... He would rather grind the opposition down with seam and defensive fields than set attacking fields for the spinners and buy an opening. Probably he declared late in the first test because he didn't trust Rashid to bowl them out, this happened last winter too, didn't it?
Nasser Hussain and Michael Vaughan were great, incisive captains, Strauss less so. Maybe Root will redress the balance. I just hope that Cook will be happy to go back to being rank and file, it would be a shame if his career came to a premature end. He is a truly great batsman.