Keep in mind, I may be a bit bias though. My Dad is a die hard Broncos fan so I watched a lot of Cutler from the beginning. I don't remember a specific example and it's not important enough for me to do research on, but all the knocks on him since he's been a Bear were present when he was a Bronco. He often seemed detached, apathetic, and selfish. Sure, he may have been the best team mate in the world on the sidelines, but us fans at home would like to see more out of our starting QB. He's not obligated to appease fans, but it would be nice.
Often a reputation is developed for a reason. Yeah, it might be unfair sometimes, but there is likely some truth to it. Ultimately, it doesn't matter what the media or fans think. As long as he has the support of the organization, he can act however he wants.