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.
Yeah. I guess I don't see it. In Nero's case, it really boils down to whether or not Kirk believed Nero could somehow still be a threat. It's a judgement call, at best.
After all, if someone believes he showed a reckless disregard for life, then why did he offer to try to save Nero and his crew in the first place? It may have been a perfunctory offer, but he still made it and would've been committed to it if Nero had accepted. That is, unless after Nero accepted, he said, "Hah! FU, Nero! I was just toying with you. I'm not going to save your sorry ass. Die, you bastard! Mr. Sulu, fire everything!" Now that would be reckless disregard for life.
That's not to say this younger Kirk isn't reckless. General recklessnes is part of the reason for the lecture from Pike, I'm sure.
I'd also think that someone with a reckless disregard for life would give signals and be singled out by Starfleet psychologists as unsuited for command.