I am getting the feeling you don’t grasp grown up concepts like business. I am about to hit you with some big words, you may want a dictionary handy.
An NFL game is not a public event. It is a franchise sponsored event held in a privately owned facility and governed under the terms of the franchise agreement. Filming rights and use of broadcast are owned explicitly by the NFL. It’s called a Copyright.
Belichick is an employee of a franchisee of the NFL. As such, his actions during franchise sponsored events are governed by the terms of the franchise agreement.
The franchisor, the NFL, gets to tell him what he can and cannot do while at an event. If they say he can’t do it, he will be in violation of his contract. The NFL can choose the remedy for the violation per the franchise agreement.
Let’s see, let’s play a little word game. Violating his contract, breaking a rule, not the same as cheating, see? Is that concept simple enough for you?