First of all I don't think any of those names will even hit the market. They will either get extended or franchise tagged. Second, I've learned with Ballard that he has a price and he won't go over even for the players he really likes. He's repeatedly refused to enter into bidding wars in free agency and I don't really see that changing now. IMO he will target undervalued players like Autry/Ebron/etc. At this point I've been disillusioned with the idea that we are going to go hard after the premier free agents if it means having to outbid numerous other teams. It's just not going to happen.
Officially? Bill Belichick is doing it himself.
Unofficially? I have no proof of this but I strongly suspect Steven Belichick, Bill's son and the current Patriots safeties coach, is stepping up and making at least some of the decisions.
The younger Belichick has been coaching the Patriots secondary for about 5-6 years now and he has 3 Superbowl rings. He's about due for increased responsibilities anyway. I'm honestly really surprised someone hasn't tried to hire the younger Belichick away with promise of more responsibility elsewhere
So yeah, Bill is taking the public exposure, but I strongly suspect that Steve is doing a lot of the real work.
Every situation is different. Other top drafted QBs include Mahomes, Jared Goff, Carson Wentz, Deshaun Watson, Josh Allen, and Lamar Jackson. You're only looking at the busts, but that doesn't mean every top QB is a bust. Many different reasons:
Mayfield - bad coaching, no protection
Mariota - overrated coming out of college, bad coaching
Winston - simply not that good, turnover machine
Darnold - was actually playing well toward the end of last year, but the current roster and coaching staff are horrible
Rosen - I think he has some talent (not franchise QB level by any means), but he's been stuck in two awful situations with tanking teams
Trubisky - overrated coming out of college, not that good
One interesting thing to note is that all the good QBs I mentioned have very good coaches with them, while a lot of the bad QBs you listed have bad coaches, so perhaps that's a big factor.