I've changed my approach and opinion on the point. I stopped listening to people sculpt how they approached fumbles and just went to the raw data. When you do that you find that New England has been, game by game, within a range of the top teams, as you'd expect, but the difference isn't that remarkable and they don't top the chart year to year.
Looking at Team Rankings data and a sample every two years from 07 and taking fumbles per game, raw.
07: NE .8, Ind .9, SD 1.1, Jack 1.1, Atl 1.1
09: Ind .7, Den .9, SD 1.0, NE 1.1, Atl 1.2
11: NO .6, Atl. .7, Car .8, NE .9, Gia 1.0
13: NE didn't make the top five
15: Car .8, Bal .8, NE .9, KC .9, Den/Pit 1.0