I think Herm/Bradway was more of the normal power structure you see in the NFL. There were players that Herm was going to recommend, primarily in free agency, that Bradway would agree to sign. I think thats normal to get a few of "your guys" in over time. Herm clearly had input but the final say went to the GM.
In Manginis case I think there was a committee structure. Mangini has say. Tannenbaum had say. Bradway had say. Clinkscales had say. And so on. I think there was alot of consensus building and I think Mangini probably made his points come across very well to help get the guys he most wanted.
I think Rex began similar to Mangini. Consensus building and a tip of the hat to his guys (Bart Scott, Trevor Pryce, Jim Leonhard). As they got successful I think Rex got a louder voice in the room. In part its because I think he was the one guy that actually became loved by the owner. Woody saw the attention Rex brought, specifically via Hard Knocks and saw that as being a huge selling point for his team. So his power grew and guys began to capitulate to him because they knew he had Woodys ear. Thats more or less why Clinkscales left.
Rex does have way more power than the other guys. I mean when was the last time you heard of a team that fired their GM, offensive coordinator, QB coach, and strength coach, and was more than happy to see their defensive coordinator and linebacker coach leave but at the same time maintain the head coach? Never. In a house cleaning like that the coach usually goes as well.