Actually, I think USUALLY some team in the background "runs the country" for every President, with some exceptions.
I've noted before my experience with VPs where I worked. They get presented a story with only one decision to be made, because the presentation is shaped so that only one outcome is obvious. VPs HATE to have to make tough decisions on their own.
I strongly suspect the "team" around a President collude to make decisions and then present the story to the Man such that only one decision is obvious, and he "makes" it. Groupthink is another variation of this.
And think about what decisions a President really has to make? He gets around 4,000 "appointments". He has no clue who most of these folks are. He can sign, or not sign, some bill from Congress. It's generally obvious he's going to sign it, it's rare a President doesn't. And then it's a political decision most of the time. What else does he do? He can order our troops hither and yon, usually with bad results in recent history. But many Presidents couldn't fine Iraq on a map, in my view.
If you get a President who is a kind of "technocrat" like Jimmy Carter he may get more involved in details, but that didn't work out so great back then.
A President is a "Show Pony", he comes out and gives speeches (Obama) and remarks to the media which are written by someone else to sound good. Then he goes back to the WH or CD and "shoots skeet" or layups or works on his Final Four guesses and watches ESPN.