Of course Manning is the GOAT.
I've never really understood why quarterbacks are judged by different metrics than any other position in football. Football players should be judged by what they themselves accomplish on the field, and nothing more. That's the only way that makes sense. Using the "rings" argument, you can change 2-3 plays in Brady's career (or Manning's for that matter) that occurred with neither of them on the field, and that would cause some so-called pundits to elevate or demote Manning / Brady in the GOAT discussion. Which is ridiculous.
For instance, just having Vinatieri miss one of his many game winning kicks (pulling a Vanderjagt, if you will), having the Seahawks run the ball on the goal line, and having the Atlanta Falcons offense take a knee the entire second half (all events that very probably could have occurred) and you have Brady sitting at 2 rings. Sad thing is, we all know that Brady's standing in the eyes of the media as the "GOAT" would change drastically in this scenario, even though he himself would have done nothing at all differently on the football field.
Brady is obviously a great quarterback, but it is undeniable that he has gotten incredible support from his coaching staff (arguably the greatest coach of all time), front office, and supporting players who have come through clutch in big moments, time after time, with a great deal of luck thrown in for good measure. Those early 2000s Patriots defenses were just stifling, and were loaded with talent.
Manning had none of those advantages. Manning had a veritable clown car of support for many years of his career, and it's hard to pin most of his playoff losses on him after those early run-ins with the Patriots. When I think of Manning losses in the playoffs, I think about the 07 Chargers loss (INT off of Kenton Kieth's hands at the goal line + defense allowing Billy Voleck to march down the field), 08 Chargers loss (occurred in OT without Manning getting the ball), '10 Jets loss (Manning puts team up with ~1 minute left, but special teams + Caldwell timeout snatch defeat from the jaws of victory), and the '12 Ravens loss (Hail Flacco).
All of this to say, if you actually take the time to consider what both players physically accomplished on the field, it's hard to argue against Manning as the GOAT. By almost any metric (advanced or otherwise) but the "rings" argument, Manning performed better and added more value to his team than Brady.