There are a number of factors at work IMO.
1) The team may completely overhaul this roster, and it may not be one that a quality coach will want to inherit. Ditching Sanchez, Holmes, Scott, Pace, Cromartie, Tebow and more are on horizon. So is potentially trading Revis. The team may well be better having a coaching search when we are rebuilding, and not during the Demo.
2) The new GM should be given the chance to keep Ryan or Fire him.
3) Woody is freaking cheap, and not wise. As I've said previously, he bankrolled millions to the campaigns of McCain and Romney. Two terribly runs campaigns, regardless of ones politics. These ego guys just don't know when to shut up. A simple statement that the new GM will make all decisions on the coaching staff would have been appropriate. A large part of me thinks Woody just wants out of Rex contracts, and is hoping he quits while he twists in the wind. Either that, or he clueless.
4) Rex may quit when he sees what a real GM is going to do with the Schlock he and Tanny put together, or he's going to have the opportunity to turn his career around by coaching up a low $$ roster, as much of the cap will be spent on players not on the sideline or field.
The Jets aren't going anywhere next season. They have to put themselves in position to get the right coach.