I'm not in favor of seeing Rex get fired, but...
1. Rex's first draft pick ever as HC was Mark Sanchez. Mark is Rex's guy.
2. Rex played a big role in bringing Sparano here to run the ground and pound.
3. Rex has always "deferred" a lot of offensive decisions to the offensive coach, even if he has recently gotten "more involved on offense."
I think part of the problem is simply the degradation of the running game over time. In 2009-2010 we had Thomas Jones and a young Shonn Greene running behind FB Tony Richardson and an Oline that included Damien Woody and a younger Brandon Moore.
By 2010-2011, I think the running game was even better with LT adding a solid veteran presence in place of TJ and we still had Richardson and Woody.
By last year the key pieces were all gone. Shonn Greene (who is a less than average starter) was running behind a Oline that had lost solid veterans and a major downgrade at FB from Tony Richardson. That was the major drop-off in the Jets ground and pound. In Rex's first two years he got to AFC Championships with a young, average QB simply because he had some weapons (Edwards, Holmes, a healthy Keller) but also had Tony Richardson, Damien Woody, etc. blocking for some decent running backs.
The Jets let the Oline degrade and were never able to effectively replace Tony Richardson at FB. They also tried to get by on the cheap at WR by going from Edwards to Plaxico to a rookie from 2010 to 2011 to 2012.
Sanchez has been really bad this year, no doubt about it, but the entire offense has been neglected since 2010 in my opinion.