I like this thread because I do think you have to isolate skill sets from stats and rings. Performance obviously counts, but we've already discussed how much better Brady is against better defenses with less talent around him to work with. Even performance is tough to differentiate or quantify because you have to factor in all kinds of variables (none of which include who has a better defense, unless Richard Seymour or Dwight Freeney are out there catching deep outs).
Manning's stats are largely the product of all the first round picks supporting him on offense.
It's also hard to separate rings from the overall team accomplishment. In the salary cap era, for better or worse, QBs are easily the most valuable players on their teams. That's really what separates the teams today, much moreso than management or even coaching. but the ring is still a team accomplishment. A great QB on a lousy team isn't going to win a ring in any era. Tom actually got the closest with the 2001 Pats. A team ranked by SI as 31st out of 31 teams before they started 0-2, lost their best player, and started a nobody at QB
I've always thought Tom had a better arm.
Even before he had downfield weapons, it still showed up in the intermediate passes. Tom's more effortless and smoother because his mechanics are flawless and he holds a lot in reserve. Bad weather QBs need reserves of arm strength. Peyton likes to gun the ball to his receivers by default, doesn't hold much back on the short and intermediate passes. A great pure passer has great touch and what makes Tom's arm so unique is the combination of touch and strength.
Tom objectively throws a better ball (i.e. tighter and more consistent spiral). Regardless of the conditions. Wind doesn't make his ball wobble.
Theisman once called it the "heaviest" ball in the game, which means it packs more of a punch when traveling at the same velocity of throws by other QBs.
Peyton looks constipated when he throws the ball. His footwork is not good and his mechanics are unique to him but I wouldn't teach a kid to copy him that's for sure.
Most players and pundits concede better field vision and pocket presence to Brady so I'll leave that alone. I would also add footwork and mechanics into the equation. Are those physical skills or intangibles or both? You could argue either way, but they're essential qualities.
...I'm not going to attach supporting documents here but I can't think of a single physical advantage that Manning has, and yet, as you say, the media tends to overwhelmingly concede the "skill" to Manning without really backing it up.