Look at the situation logically. When a new GM is hired they want to hire their coach and bring in a QB. Idzik didn't have the opportunity to bring in a HC, he drafted a QB. He wants that QB to play. sanchez never won a game for Idzik, he has no loyalty to him but he has for Rex and Rex has been loyal to him.
Why hire a new GM in the first place if everything was going so well? If Rex Ryan was as good as you make it seem he is the GM wouldn't want to can him (or the QB).