The city shown in the film was presumably ShiKahr
How so? It's supposed to be the city with the Vulcan Science Academy in it, after all - and nothing indicates that Spock's hometown would be the location of this exalted body.
Was it Space Seed or Tomorrow Is Yesterday where Spock said that the records from the 1990's/2000's era were "fragmentary"?
"Space Seed" - but only in the context of Spock being unable to dig up the exact shipping records that would have revealed the identity, mission and flight plan of the Botany Bay
. Nothing about that statement suggests that mankind would have forgotten who was the president of Guatemala at the time, or what children a nobody Captain from the USAF might have borne. Details of Khan's career and ancestry would probably still be available in full.