Okay, but if Idzik's job is to asses all aspects of the Jets, including Rex, through training camp and games, then you are basically saying that Woody gave Rex the job for at least this year. Because I don't think Woody has in mind Idzik saying after preseason game #2, "We need a new head coach now, let's start looking".
If Woody gave Idzik the right to fire Rex this year, Idzik or any other general manager would do so and move his own choice in, guaranteed. That's why, if Woody was willing to let the new GM get rid of Rex, he should have cleared Rex out with Tannenbaum and let the man hook up with another team while the positions are available. To do anything else would show lack of character.
That is not guaranteed at all. It is entirely possible he would keep Rex on even if he was allowed to fire him. You are making some grand assumptions, such as a GM forming the same opinions that you have with Rex, and also that Idzik had someone in mind - "his own choice" - who was presently available and who wanted to take over a train wreck of a roster in 2013. You're looking at the roster situation with July's viewpoint instead of January's.
Coaching this team will be far more attractive, for a new coach, in 2014 than in 2013.
Rex, while he did plenty of boneheaded things, also had enough cards stacked against him that 6-10 might not even be considered a bad season around the league:
- No QB
- No backup or future QB
- Injured #1 WR
- Unready and then injured #2 WR
- 1 healthy slot WR
- starting TE was injured
- RBs underwhelming
- FB awful
- While the OL graded out ok comparatively, the stud LT and C having down seasons while the other 3 ranged from meh to decent. RT was an undrafted castoff who was forced into starting. Roster's other linemen included a 2nd round bust the team drafted and a #2 overall pick bust that St Louis just wanted to be rid of.
- Top player on defense, Revis, got concussed during week 1, causing him to miss week 2, then got knocked out for the season during week 3. In other words, for the whole season he played in 2 games and got knocked out of both.
- Team's only player with obvious, potentially star-like pass-rushing skills was a rookie who doesn't have an obvious position on a team chiefly employing a 3-4 front alignment.
- 3 starting LBers were years past their prime. The 4th starting LB took a major step backwards and guaranteed $25M from 2012-2013.
- Recently re-signed NT was playing injured & play dropped off some as well at age 33. Player drafted to take his place merely showed "some flashes" in 2012 (and was injured for the middle third of the season anyway).
- OC was garbage (yes, he was Rex's hire, but it still was a handicap. Particularly for a HC whose expertise is on the other side of the ball).
- Special teams went down the toilet after being perpetually among the league's best for a decade.
So you will have to accept that there are GMs in the NFL who would look at all that and say to themselves that 2 games shy of .500 isn't the worst thing they could imagine as a final record. A GM could easily conclude that, given that the team went to the AFCCG in 2 of the prior 3 seasons before, that the most recent 6-10 record might not be due to poor coaching from Rex.
And a GM could say that Rex is a big part of the failure and needs to be tossed out immediately as part of a culture change. But there is no "guarantee" that every 1st-time GM would make that move if given the opportunity.