I always saw the "Let them die!" line as an emotional outburst, not a statement of policy. And certainly Kirk proves by the end of the movie that, ultimately, he's capable of looking beyond his own history and emotional demons--just as, indeed, he did in "Day of Dove."
People are complicated. I'm sure that, back during the Cold War, that there were plenty of American soldiers that hated the Russians in the abstract, but we're still capable of seeing the humanity of the enemy as well.
Kirk isn't always right all the time, and was certainly capable of letting his emotions get the better of him on occasion, but his higher angels always prevail in the end.