I'm trying to remember, when did this little tradition actually start? I always thought it was more of a Belichick thing, since he had a lot of success with NYG before going to Cleveland (ugh!)
Coaches and the clubs used to have a lot more say in scheduling the preseason games. When Parcells took over in NE and Mike Holmgren was hired by Green Bay, they played each other in preseason twice, and then in Carroll's first year. But then the league stepped in and took a more active role after two "no-nos" happened, and both in the same preseason: Two teams played each other twice in the same preseason (Dallas and Oakland) and there was a game between divisional rivals (San Diego and again, Oakland).
But yeah I'm sure a lot of it has to do with the end of the preseason. By that point everyone just wants to get it over with.