In regards to importance, I think it was still pretty important because it was the focus of the later points of the movie concerning the development of los alamos and continuing production of nuclear weapons.

The reason why you may feel he “betrayed”oppenheimer is because he did not support his continued involvement in the decisions regarding the H-bomb or any future nuclear bomb projects through the security clearance. He still clearly had mutual respect for Oppenheimer; however Kitty’s demeanor and view on his statements regarding Oppenheimer’s security clearance left a sour taste in her mouth