...? No, Burnham was a person recovering from trauma building back up from losing almost everything.
And, it's depth was just fine. It is actually quite Shakespearean as it exists in an "elevated reality" with intense soliloquies about emotions, like love, death, revenge and guilty, among others.
Also, no. Kirk was a soldier, and leader. He took care of his people, was harsh and unrelenting against his enemies, and gave digs in to his subordinates at times. Not a cowboy, as much as pop culture would like us to believe.