I disagree. You mentioned Brees and Roethlisberger (who went through a spell without PO's when the roster took a downturn). Philip Rivers is a darn good QB who has been sabotaged by his GM's. Romo couldn't get there for years with what Jerruh was doing to the club. Dan Marino (who, for my money, is the GOAT) missed the playoffs seven times in his career. The Colts have missed in back to back years with Luck. The Falcons went through a three year drought with Ryan.
Look, I'm not saying any of the above (other than Marino) are as good as Rodgers, but they're all somewhere on the spectrum of "great" to "elite" and their teams still missed because the rest of the roster was mismanaged. Their GM's were not able to just stick them in there and go on autopilot and make the playoffs (which is how several people characterize TT's job).
I realize the defense has been an issue and must improve. But you can't say that Rodgers is relied upon more heavily than any QB in the league. He has a top 5 offensive line-- according to PFF they are the best pass blocking unit in the game in terms of pressures allowed. And Rodgers holds the ball longer than anyone not named Tyrod Taylor. He also perennially has one of the better groups of pass catchers to throw to. This takes nothing away from him. He's amazing. But TT has also done a pretty good job of supporting him and keeping the roster healthy overall over his time as the GM.
If you want to say that TT is derelict in not signing FA's to help the defense, I think that's fair. But to say he's just been able to draft Rodgers and coast is nonsense.