Khan may have been playing the victim card, but I never saw any indication that Kirk was buying into that.
Of course not. He's the same guy who blew up the Kelvin Memorial Archive and killed Admiral Pike.
But Kirk doesn't know him beyond being a highly effective and murderous super-soldier. He could have just as easily said "My name... is Master Chief
!" and it wouldn't had made a lot of difference. The point being Kirk -- and a portion of the audience -- doesn't know what WE know. They don't know what "Khan" means in the context of Star Trek, and they don't know that Khan is actually a much bigger threat than Marcus could ever hope to be.
Which is how Khan winds up besting them in the end. Despite his very reasonable suspicion that Khan could not be trusted (and having Scotty stun him in the end), he was so focused on taking down Marcus that he under-estimated Khan and let him outmaneuver him.
I don't think that all plays out very well, especially for newer viewers since the film never actually explains that Khan was one of Earth's worst tyrants.
I don't see how that's even relevant, especially since anyone who knows how to google would be able to figure that out inside of fifteen seconds (and the more curious viewers would find the title "Space Seed" in their google results and Netflix the entire episode).
It doesn't matter who Khan was in the 21st century. What matters is who Khan was in the Enterprise's timeline. This is why Spock SPECIFICALLY asks OldSpock "Did you ever encounter a man named Khan?" And OldSpock doesn't waste his time with historical background, he tells him exactly what he (and therefore, the audience) needs to know.
Put that another way: when a young fan asks a vet "Was Khan a villain in the old Trek?" the answer will usually be some variation on, "OMG, you don't know? Khan was the BADDEST viallin of the old Trek! He had a whole movie and everything! He even... you know what? Fuck it, we're goin to Best Buy! Today's your lucky day."
Maybe it just didn't work for me like it did for you, because I was rolling my eyes too much over the phone conversation, and how everyone seemed to not be a little curious about why Spock is suddenly speaking to his future self, unless he's been telling everyone about it before the movie started.
There was a deleted scene in STXI where Kirk
told everyone about FutureSpock as his explanation for how he got on board the ship (which is basically why they let him stay in charge). Apart from that, there's indications in the comics tie-ins that Kirk and Spock both went and explained it all to the crew afterwards, especially since OldSpock is apparently a pretty big figure in the New Vulcan settlement and hasn't bothered to use an alias.