Funny, I didn't find it particularly bloodthirsty behavior by Kirk, at all. Time was of the essence, and the obstacle needed to be removed quickly. Remember, the scene cuts back and forth between the battle and the assassin preparing to shoot as the president gives his speech.
I don't find either action particularly blood-thirsty either.
But you have some folks trying to define the new version of the character as having a reckless disregard for life. They were both in situations where they could've called off the dogs but didn't.