Because Rex made a very strong start his first two years. The team lost focus the last two. After reviewing the situation, Woody decided that Rex deserved another chance with a different GM, then set about to find another GM to work with Rex.
Besides, Tanny had already worked with Mangini and survived Mangini's firing. Now Tanny works with Rex and after a good start things go south again. This time it was Tannenbaum's turn to go.
Woody's decision to keep Rex for now may well have prevented some prime GM candidates from taking the job, since GM's like to select their own coach. This is balanced, in Woody's mind, by the fact that Rex can be an outstanding coach and he did not want to let him go only to see him blossom with another team. When push comes to shove, great coaches are harder to find than good general managers.
That doesn't mean that Woody is keeping Rex forever, it only means that Woody has decided to give him another chance, based on his strong first two years.
Woody pays the bills, and decided he was going to do it. Besides, in any organization major changes should be swift and decisive. If you're going to replace Tanny, the new man will likely want his own coach, and the chances that the coach he would choose is Rex is small. So if you are going to bring in a new GM and give him that choice, Woody might as well send Rex packing along with Tanny. To fire one of the two major factors, then keep the other dangling for several weeks and then let him go then, (which would prevent Rex from lining up another job in the peak hiring season right after the season ends), would be weak, vacillating, and totally wussburger decision making.
Woody decided to give Rex another chance, and set about to find a qualified GM to work with him. Simple as that.
I guess we just see this totally differently all the way around. Yes, Rex started strong, but the team going south is as much on Rex as it is Tanny. Tanny didn't turn the team into a circus, make all the stupid predictions and comments in press conferences, didn't lose the locker room, didn't wuss out in games settling for FGs instead of going for TDs, and all the other crap that Rex has done. Woody isn't qualified to make any football-related decisions, especially regarding the capabilites of a HC and CS. His ego is in the way. Yes, he pays the bills and has the right to do whatever he wants, but unless he wants to look like a dumb jackass, alienate his fanbase, and waste a lot of money, he needs to realize that he doesn't know what the heck he's doing and look for the best GM he can find, then step back and let that GM do the job as he sees fit.
Making a decision to keep Rex, and thus knocking out perhaps the best, most qualified GM candidates is short sighted at the very least, and moronic at worst imo.
Unless the first thing the prospective GM candidates asked Woody was whether they could fire Rex, or Woody asked them (or had the executive search firm ask), then he might not have known what they would have done since they declined to even interview. The new man could have decided to keep or fire Rex. That was part of the reason for hiring a new GM who was actually qualified and knew what the heck he was doing. Hiring the best candidate and then letting him decide is not being a wussy. When the owner of a business hires a new manager or President, the owner doesn't automatically fire his whole staff thinking the new manager will want to bring in his own staff. He lets that manager make the decisions on who goes and who stays. The same thing happens in sports. It is rare that both the GM and HC are fired. To say that he might as well have gone ahead and fired Rex too is ridiculous and has nothing to do with reality imo. There might have been candidates who wanted to keep Rex, and then that would knock them out. The bottom line is that the HC and CS should be the GM's decision, not the owner's.
I'm also not sure that I would agree that it's harder to find great coaches than great GMs. I know that you said "good" GMs, but one ought to be looking for the best leader of the franchise one can get (GM) and not just someone who is good. Coaches come and go, but if you have a great GM who builds a strong foundation and organization, and keeps the team stocked with great talent, finding a good coach who is good enough to win with that talent shouldn't be that difficult. Being in the major media market in the country, with an owner who has deep pockets and is willing to spend, and having a great, respected organization that will keep the team well stocked with talent would have HC candidates salivating at the prospect of coming to the Jets. They would be seen as saviors and it wouldn't take much at all to become the greatest coach in franchise history.
It may be as simple as you say, and it certainly appears to be, but that is why I despise Woody and don't really trust that this team will ever get it right as long as he is the owner. You spend the money to hire an executive search firm to find the best candidate they can, and then hinder them in their search by telling them they have to find someone who agrees with you and will be your puppet. Absolutely egomaniacal and moronic. If he doesn't wake up and soon, I hope the bastid goes bankrupt and is forced to sell the franchise.