I appreciate your intent, but there are a couple of issues with this.
First, teaching mobility is similar to teaching speed. You can tweak technique a little bit, but you can't make any significant improvements. Either you're Petras or you're Beathard. You recruit mobility, you don't teach it, not even at KF's little ole developmental program. LOL
Second, Tate is a fiery fella who didn't cotton to being a game manager and being placed in the KF straight jacket as a player. He'd never fit on the KF staff or, if he did, he'd be neutered and be of no use whatsoever.
Iowa needs offensive coaches like Drew Tate, but they won't get any until KF rides off into the sunset or is six feet under, whichever comes first.
I'm interested to see where his coaching career leads. Like all people, he's matured from his college days based on the podcast interview he did with Long and Kakert. Realized in Canada there was still stuff to know/learn about football, and he was taking the humble approach to coaching football. That will serve him well.
To your point about mobility and speed. Yes I would agree. QB's don't magically get tons better pocket presence or moxie or whatever you want to call it. First time Tate went into a game in 2003 as Chandler's backup as a freshman he moved around in the pocket with ease and found ways to get a throwing lane with eyes downfield. Yes, that stuff can be drilled and improved to an extent, but it's just near impossible to take a guy with zero pocket presence or escapability and make him really good at it.
The plays Tate was making in 2004, such as the Wisconsin game where he reverse pirouettes, sets his feet and hits Solomon 50 yards downfield for a TD, that is not taught. That is a guy who senses pressure, has the athletic ability to get away, doesn't panic and in a split second find an open receiver downfield. Recruit guys that can do that, and coaches look a lot smarter.
As you pointed out, Beathard had it. Once he got to the point where could lead the offense, he was the right choice as QB. And magically the Greg Davis offense was good enough to go 12-2 in 2015.