I agree with that course of action. Unfortunately, the Patriots have tried it in the past and the league has shown that they won't make a big deal about infractions by other teams. While not a formal complaint, the Patriots did eject a Jets camera crew from an unauthorized location in 2006. The league took no action. In 2009, they did formally complain about a Jets employee tampering with footballs. The NFL suspended that employee for an unknown amount of time, took no action against the team or any players, and buried the incident so deep, no one heard about it for 6 years. Last year, the Jets owner tampered with Darrelle Revis. Even though the tampering occurred in public, was almost a word for word recitation of the example of tampering from the NFL rule book, the Jets succeeded in signing Revis, and the organization has a history of tampering, the league chose to hand down a punishment that was lighter than the normal penalty for that infraction.
I wish we had more information about all of these incidents. I can almost understand not making a big deal publicly about the league ignoring infractions by other teams or letting them off lightly. What I don't get is the apparent thinking that they (the Patriots) would also get the "slap on the wrist" treatment. The NFL's pattern of behavior has been the opposite of that.
Having said all that, there's no reason to stop filing legitimate complaints, if only to show the league's inconsistent punishments and double standard.