Remember that a lot of FAs are now at that point where if the Patriots sign them; they are being told that they were gotten "for a steal", and the Patriots are notorious for being ruthless when it comes to loyalty (even Tom Brady's dad expects the Patriots to be dicks to Brady when they want to move on from him).
If two teams wanted to hire you, and one of them that is successful is known for underpaying their employees equivalent to their worth, and the other one is not successful, but pays employees what they are worth or even more...which one would you want to go to? The successful but get paid much less company who is known for being ruthless to their employees, or the not as successful company that pays you significantly more and might not be as ruthless. If you want to maximize the amount of money you make, the Patriots aren't really ever going to be your team.