Tebow "chose" NY because he was lied to by the Jets brain trust in that he would have a fair shot to start. Teebs and Rex apparently share the same agent so he believed what he was told. It was clear that only the Jags owner wanted him, so NY was what he thought was a better choice.
Also, don't forget that the Jets already had a back up signed that they promised would be the clear back up( I forget his name). They had to trade him somewhere to placate him.
So clearly the jets FO has a credibility issue when it comes to player promises.