Great question. My theory is---Bill Belichick got his offensive inspiration from the Pacman video game---actually I am being somewhat serious in an analogous kind of way---the best way to beat the ghosts was to escape them up and around the middle, which is exactly what BB does with his offense.
Think of it: where (typically) are the worst pass defenders? In the middle.
What is the shortest distance between two points? Over the middle.
So what he does is, he acquires and drafts big target TEs to threaten the seams, and then shifty, quick footed slot WRs to exploit mismatches in the middle---and if you take those two options away, here comes the quick RB as the third pass option over the middle---thus it is three waves: TEs, slot WRs, speed RB.
Every year since Randy Moss left town the media speculates that the Patriots need to draft a #1 WR...and every year they don't. Why? The perimeter is not their priority.
It's an ingenious strategy too for trying to preserve a QB in his late 30s. The ball quite often comes out very quickly and are mostly 10-12 yard flicks of the wrist.
Sort of a modified west coast offence?