I think the core reason behind John Schneider's success has been his moneyball-type philosophy, both in free agency and the draft. In short, you are better getting a $5 value for $1 dollar than you are getting a $50 value at $46 dollars. His mastery of finding value acquisitions big and small from low resource expenditures is the driving force behind Seattle's incredible roster efficiency. A critical part of it has been the help he's received from a stellar coaching staff and what is probably the best locker room dynamic in the NFL.
I agree that in the big picture, Tom Brady was not 300 times more valuable than Peyton Manning just because his #199 pick was worth 1/300th a #1 pick (on the Jimmy Johnson chart). I totally agree on that point. At the same time, Russell Wilson does bring insane value to the franchise not only because he cost less in draft pick capital, but also because his contract is 1/40th what similar QBs around the league are making. So I do think there is a lot of value in getting great players on cheap contracts- though these days even 1st rounders are on cheap contracts.
I agree completely on the points you made regarding the arbitrary "big board" that casual fans use to judge value, as if all 32 teams painstaking track Mel Kiper's best available. That said, I do think there is always a point that serves as the floor for every single player, and some GMs are better at collecting intelligence on where those places are than others. John Schneider is fantastic at it, and it was a big reason we got Russell Wilson in round 3 last year.
In summation, I think an excellent drafting GM must have the following traits:
#1: He must be well connected to his coaching staff. The more connected, the better. Establishing a good relationship with the players helps too.
#2: He must be a nuanced talent evaluator that draws his own conclusions intelligently and is immune to mindless groupthink or hype. Working long hours helps too.
#3: He has a "6th sense" for knowing when players are about to leave the board, which will often be at odds with what the media is projecting.
#4: He should probably trade down a lot more than he trades up, and draft for upside as much as possible. In baseball, the most valuable players are typically the ones who play every day while hitting for a high OPS (and since SLUG% carries OPS a lot more than OBP does, the best players often hit a lot of homeruns). In this analogy, trading down is a bit like drawing a walk (OBP), and swinging for high upside players is like swinging for extra bases (SLUG). Guys that hack and hack while drawing singles usually don't post great WAR numbers, unless they also have fantastic defense or some other compensating factor. I feel it's the same way with NFL GMs- Tim Ruskell being the perfect example of the no-walk singles hitter.
#5: He must have a plan and a structure, preferably one that is on the visionary side of the spectrum. He must build the kind of fertile environment that risky prospects can succeed in.
#6: He must be humble, listen to others, and have the intestinal fortitude to avoid panic decisions.
We are very blessed to have our GM, as I feel he passes every one of these criteria.