The characters were pretty on point outside of a few key differences. Kirk was a rebel not the pretty disciplined well polished officer that only bent rules when he knew it was right to. This was a rebel Kirk.
You have to consider the impact of the opening scene. He lost his father before he ever knew him. Many classic time travel tales spoke of the huge impact you could make on history if you did the smallest thing. In this case, we have a futuristic space ship from a long forgotten race (connected to Spock) who kills the father of a future hero of the federation. In the first few days of this timeline, Spock loses his mother and his home.
I think you could explain a more rebellious Kirk and a more emotional Spock based on these events.