But after missing the boat on a Clay Mathews and a few other picks I hope we go for a game changing player..
You do get that Belichick & Co. don't know, or even THINK, that they are moving away from "game-changing players", right? It's not like he's sliding back going "well, I could have this Pro Bowler, but instead I want JAG plus an extra 5th rounder".
Instead, they look at their board, and they determine either that they have a bunch of guys with very similar grades, or someone with a higher grade that is very much NOT likely to be taken before they go again, and then they slide back a few slots, pick up "extra value" and then get the guy they wanted all along, or another guy with the same grade.
Let me give you an example: You have the #1 draft in the proverbial all-time NFL draft. Any player, any era. You can take Montana, Brady, Manning, Elway, Unitas, Jim Brown, Butkus, etc. etc.
The guy with the #3 pick calls you up and says "I'll give you my first and third round pick for you #1. Do you take that deal? Of course you do. Why? Because is there really THAT much of a difference between whichever QB you want to take at the top of the draft board? No, not really.
That's what Belichick does every time he's sliding back. Getting more vlaue than the difference between the players available NOW, and the players likely to be available when he drafts again.