We don't know how old these characters are supposed to be.
This movie is supposed to take place about 6 months after the last one.
They could've skipped introducing Khan, played out that history off-screen, and have him a part of a plot, not the centerpiece of the story. If my instinct is right and this is going to be about Captain Kirk's mistakes, it could be a culmination of several missions that come back to bite him.
I suppose they could throw out a line saying something like "Hey, Jim, remember that 20th century sleeper ship we found two months ago and handed over to Starfleet instead of thawing them out ourselves? You'll never *guess* what's happened since!..." But that would be lame and contrived.
I agree with your last point about Kirk's "mistakes," but I would be disappointed if Khan somehow figured into them at this point.