From all of the accounts I've seen, heard, and read: he's very intense and demanding. But yes, he's taciturn in regards to his emotions and how he personally feels about the team. That can lead to fans viewing him as a prevaricator.
He's the anti-Jim Mora in a way.
In the end, I'm not so sure it will be his call to make on Bradley. I think if Bradley is gone, it will be Chris removing him. This is his final chance to keep his job in the eyes of many, so he's not going to squander his opportunity by putting his future in Bradley.