Elite has nothing to do with whether a QB is a game manager, gunslinger, or improviser. Elite is used to say you are good at whatever type game you play.
All of then are game manager to a great degree. Some due to mindset, some due to system.
Crunch time is going to tell you what type QB you have.
Does he take the safe throw, the check down, the dump off and wait for the next play? (game manager)
Does he stand in and force the ball deep trying to make a play? (gunslinger)
Does he run around keeping his eyes downfield trying to make the best play he can? (improviser)
A game manager needs a superior run game and defense, no question about it.
The other two need a defense just good enough to keep them in the game where they have a chance to win it.
Game managers won't go halfcocked. They play it by the book. Their primary concern is not to put the team in a hole. Take care of the ball. Keep the chains moving. He's not a playmaker, he lets the playmakers make the plays. Wide receivers in this system post consistent numbers, but rarely qualify as game changers.
A gunslinger is not worried about down and distance or picks. He's fun to watch. He's trying to win. He will force the ball downfield. He's expected to be a playmaker, and the guy the team rallies around when they are down. WR with these QBs get a lot of press and accolades for the yards, touchdowns, and receptions but none of the flack for the picks and bad throws.
An improviser is not going to make any major mental mistakes but he will go downfield if the opportunity presents itself. He's a playmaker. He puts himself in the position to win and takes the best possible shots he's given. WRs are the same as with gunslinger types.
I think we all consider Drew Brees Elite. But Brees has a gunslinger mentality. He's an extremely accurate downfield passer, but not likely to challenge double coverage as, say, Aaron Rodgers would.
Tom Brady was considered a game manager a few years back and yet he has gunslinger tendencies while being a pure pocket passer. Rather than run he hits the deck and curls up around the ball (my first thought was like a baby).
Rothlisberger was an improviser in past years. He uses his feet to buy time and go downfield. Tony Romo, and of course, Rodgers.
Some QBs, (Tannenhill) don't have that quality to flip off the game manager switch. And what is Tennessee winning with? A bruising running game and defense that is shutting opponents down.
Dak needs to find that game manager switch this year. And figure out what he wants to be.