In his current position as Kansas' head coach, though, he wouldn't even recruit a quarterback like Brady.
Like many coaches in the college game nowadays, Weis is transitioning away from the traditional pro-style offense and running a more up-tempo one that features a dual-threat quarterback like Jayhawks starter Montrell Cozart.
"In this offense that we're running, I think athleticism at the quarterback position is a critical factor, and (Cozart) definitely has athleticism," Weis said at Big 12 Media Days. "Besides being able to stand in the pocket and deliver, I think that that added element to either on called runs by quarterback or just scramble ability to bail you out of some trouble situations gives us a better chance to score more points."
Weis built his reputation coaching Brady and by sending traditional drop-back quarterbacks like Brady Quinn and Jimmy Clausen to the NFL. A rough ending at Notre Dame, the forgettable season spent as Florida's offensive coordinator and a pair of stalling offenses in Lawrence have changed his thinking as to what you need at the position: it's dual-threat signal-callers or bust.
"You're playing 11-on-10 football (without one)," Weis added. "If the quarterback is never going to carry the ball and is not a threat to the defense, and they don't have to worry about him, they're plus one as far as numbers go. I think by having the quarterback being able to be one of the guys that carries the ball, puts much more stress on the defense.
"I think the true dropback quarterbacks have been exposed. And that's one of the reasons why I changed what we're doing."
Weis hired John Reagan from Rice to run Kansas' new spread offense and will be almost completely hands-off when it comes to playcalling on game day. It's a big change for somebody closely associated with some high-flying passing attacks run by a quarterback dropping back five and seven steps at a time.
So while Brady's eligiblity ran out a long time ago, he might not even have received a phone call from Weis had he been a recruit this year with the Jayhawks' newfound emphasis on the quarterback being a run threat.
Perhaps that's something for NFL traditionalists to take note of.