I don’t know why in the hell they didn’t just say that they gave “Khan” plastic surgery to disguise his identity, it would have fit nicely because he was a FAMOUS DICTATOR and no one recognizing him would be like no one recognizing HITLER so of course if they wanted to use him as secret agent they’d change his appearance because the last thing you want in secret agent is someone who’s identity can’t be kept secret. Even if they really drop the ball on teaching history in the future and no one would recognize him someone could run his picture through a computer and know instantly.
But of course Earth history expert Spock had to ask his older self who Khan was in that movie, so who knows what in God’s name they were thinking, most likely they weren’t thinking at all. Maybe the Borg went back in time and made Khan White and English and completely lacking in charm at the same time they made Zephrem Cochran much older and taller and dirtier than he was supposed to be.
I gave up trying to figure out what coherent logic went into what they did in those films. Not a shred as I can see.