But Kirk did step back and examine what he was about to do and the morality of such after seeing his officers react negatively to the mission.
Yeah, he did. I think part of what the movie was trying to do here was *flip* Kirk and Spock a bit as the story plays out, with Kirk adopting the more "by the book," selfless attitude, and Spock finally succumbing to rage because he loses his friend.
So, by the end, they understand each other better.
I liked this aspect of the movie a lot. I'm still digesting it, and I'll probably want to see it again, but ultimately I think my opinion will mostly end up being that I wish they had told the story without lifting so much directly from TWoK. That mostly fell flat for me, at least initially. But we'll see.
Uhura gets through to Spock by making the logical argument. I don't think a purely ethical argument would have worked at that point, but that is intentional, the idea is that Spock has abandoned the ethical perspective, while Kirk has adopted it.