If the bears are truly getting rid of Trestman, I think they keep Cutler. But I would not be opposed to Cutler->Griffin trade. Perhaps change of scenery would be good for both. Cutler is 31 with a big contract, so that could mean only a few more years but it could make us competitive and allow us to continue to develop Cousins (?) or draft another developmental QB. Cutler would have DJax, maybe Garcon if he's here, a decent RB, good TE...
Many issues with Cutler, as he's not playing well. But one issue is our protection. This
link and this
link suggests he holds the ball too long (~2.8 sec)-- and would get killed with our protection.
Biggest -> I don't see Cutler causing much media controversy recently, and not tweeting or arguing with fans on facebook. However, I also don't see him as a strong locker room guy / leader.