They're both masters of their craft, and when you isolate pure QB play, there's no way anyone can prove that Brady is more responsible for team success than Brees has been.
That's the difficult part. Payton is an offensive genius and so is Bill, and we've never seen Brady outside of that system.
If we're being honest, they've probably both benefited from the systems they're in. We've seen Bill have success with other QBs. We saw Brees perform at a "very good QB" level in San Diego but not hit "great" until he got together with Payton. And we also saw Payton succeed with a lesser QB in Bridgewater. Like you said, it's really hard to separate the coach and system from the player with these two. We'll learn a little bit in Tampa, but it's so late in Brady's career that we may not be able to really make any judgements from it.
I actually think the 2000s/2010s QB argument is more complex than most do. I'm not even entirely confident that if you took Aaron Rodgers or Philip Rivers and put them with Payton or Belicheck that they wouldn't have had very similar success to Brady/Brees. But, they weren't and they didn't and so it goes. In the super-passing era of the NFL (call it the last 20 years) there are three QBs head and shoulders above the rest. Brady, Brees, and Manning. That's a universal truth. "Great" is a combination of talent and luck to be in the right place in the right system at the right time. It's not always fair - just ask John Stockton or Kevin Garnett.
To the football watching world outside of Saints fans, the rings will set Brady apart. It also may not be fair, but it makes sense. It's not really much different from the Lebron/Jordan debate, honestly.