I've always found the term game manager a bit superfluous. Peyton Manning is a great game manager. Drew Brees is a great game manager. Tom Brady is a great game manager. The only distinction people make between a "game manager" and quarterbacks like these three is that they can make almost any throw. It's a ridiculous term when all of these guys are great game managers, but characteristic apparently separates them entirely.
It's one of those phrases that doesn't mean what it says, at least not as specifically as you're defining it. I agree that most times Brees and Manning are good game managers, but in the football lexicon, "game manager" means not just what those guys do, but a guy who does things safely, not aggressively, and whose primary job is to
not lose.