I think you are probably missing the point of the scene. We've seen Spock get violent or display apparently emotional behavior numerous times over the course of TOS, in Man Trap, The Galileo Seven, Amok Time and All Our Yesterdays for particular reasons. We have been made aware he has a unique relationship with Burnham and they are arguing in private. I don't see his behavior as out of character or that he's obsequious C3P0) suddenly been turned into the equivalent of a 'hardened assassin'. IMO, that comparison seems to me rather hyperbolic.