If the question is, does a coach make a great player, the answers is, of course not. As you say, a great player is motivated from within.
But the reverse is also true. Great players do not make great coaches. (IMO, of course) Stephen Jackson was a great, internally motivated player who played on lousy teams for lousy coaches. As have other great players in league history, he toiled in vain on a poor team and he could do little to raise that team's ceiling.
The thing about football is that a couple of guys cannot do much by themselves. Consider Donald last year. He had a huge year, and he certainly raised the defense's ceiling, but not enough to make it a great defense. Even great QBs can only do so much. This is very different from the NBA, where 1 or 2 truly great players can lift a team into contention, even if the coaching is mediocre.
The test of a coach is not whether he coaches great players. It's how he does with other players. Does he find ways to get production out of limited players? Can he motivate players without internal drive to do pretty well? Can he create team discipline that maximizes the odds of success, whatever the team's talent level?
To me, the proper measure of a coach is not what he wins. It's how close to its ceiling he gets a team to play, consistently, over time, and through personnel turnovers. That's the thing with Bellichek. He's had Brady, and some day soon we'll find out how much of it was Brady. But he has sustained achievement through several roster generations and turnovers. I don't think Brady accounts for that. I think Bellichek is setting a standard for being able to sustain excellence over time with hundreds of players by seeing the value of not-great players and leading them to be productive.
That's what great coaches do. A great coach may or may not have the good fortune to be blessed with championship-level talent. But his teams will always play at their ceiling and achieve more than they should. There are a lot of college and HS coaches who display consistent greatness with limited resources.
And there are fortunate coaches who win rings by being at the right place in the right team with great players. I personally don't see either Vermeil or Martz as great coaches, but their team achieved greatness when the assemblage of talent was good enough. And because they were not great coaches, they could not sustain it through personnel turnovers.
Well, I'll quit at this point. Great players are wonderful, but IMO, no NFL team can succeed without excellent coaching.