The team quality point is well taken, and I did find a -0.15 correlation between team quality and HFA. However, that's relatively small.
This is probably more exploration at some point, but I ran the numbers again since 2002 using what is probably a better. This a regression between margin of victory, strength of schedule, and home field advantage. It is analogous to Football-Reference's SRS stat (which is why I termed it the same), except while Football-Reference gives all teams the same home field advantage in the regression, I have let the HFA term fit individually to each team.
I suspect this makes it less predictive than normal SRS going forward, but by allowing the extra degree of freedom, makes it a better fit to describe historic events. These numbers are in terms of points:
This basically means, once you account for strength of schedule (which my previous FC posts did not do), the Patriots have effectively had 0
margin-of-victory home field advantage since Gillette opened.
EDIT - these numbers assume static team quality and HFA, which is obviously wrong. To be honest, I'm not sure this will have an effect, but I will check.