I'm not sure anyone has the real story, it is only conjecture at this point.
My suspicion is that since we saw Stockar McDougle being substituted early for Carey in the Buccaneer, Broncos, and now Jets games, is that the coaching staff really wants McDougle to win back his job at RT. Once that happens, I can virtually guarantee that Carey takes the starting left tackle job back from McIntosh immediately.
Right now we're fielding the McIntosh-Carey tandem because they are the most mistake-free pair we have. But their profiles are the complete opposite of what we want to be. Your right tackle should be a dominant run blocker with maybe a few weaknesses in pass protection. Your left tackle should be a dominant pass protector who can afford to be a bit pedestrian in run blocking.
But McIntosh is a much better run blocker than pass protector, and Carey is a much better pass protector than run blocker! So why not just move McIntosh over to RT? First off, it MAY have been a stipulation of his contract restructuring. Remember that we had to promise not to sign or draft anyone to compete with him? perhaps moving him to RT where he'd compete with acquisition Stockar McDougle would be a violation of that deal. We should ask Brett Tessler about that one. My other suspicion is that McIntosh, who has never played RT before, is uncomfortable with the idea and probably not ambidextrous so the coaches believe he would not be the same player at RT that he is at LT.
Ideal tackle combo is Carey-McDougle, they just need McDougle to finally get on board mentally and stop making mistakes. As soon as the coaches believe McDougle has come around, he'll be on the field. The coaches didn't make him their SOLE offensive line acquisition to have him sit the entire year.