Well, maybe AP - assuming they could somehow make the numbers work, which isn't likely - could be a strange way around the Pats' secondary issues. There are many ways to win in the NFL. You can win with big-time offense, or you can win with big-time defense, or some combination of the two. The Patriots, with Adrian Peterson as the lead RB, would have an absolutely ridiculous, off-the-charts offense. One that probably, in all honesty, could (should?) average more than 40 points a game. He's that good. The Pats have had solid, workman-like RBs during Brady's tenure, and that's been plenty good enough for them to average 30+ points a game and win 4 SBs. But they've never had, in this era, a running back like Peterson (well, how many teams HAVE had a guy like Peterson?). He is both powerful and incredibly explosive. Nothing he can't do for his position. He can grind out the tough yards. He can rip 80 yard TDs if you give him a seam. He's the kind of threat that teams MUST account for.
So you have him, Edelman, LaFell, Gronk, and Chandler on the field at the same time. What does the defense focus on? If they play man, that weakens the run D unless they stack the line and offer no help to the WRs. Then just go play-action and win so many one-on-one battles (including Gronk on a hapless defender).
Defenses play the pass and Peterson rips off huge chunks of yards.
Defenses play straight up, and the Pats can pick their matchup.
If you're going to be weaker on defense, one way to deal with that is to become so unstoppable on offense that it doesn't matter. And doing it by adding a huge running upgrade means you're making your offense more diverse - you're not just adding another receiving weapon (like Andre Johnson) to the mix. You're still able to win ball-control games or high-scoring shootouts.
Now, we all know that this won't happen. But my goodness.