Kirk, Spock, and other Starfleet officers presumably had diplomatic courses at the Academy. They're authorized to negotiate with other worlds on behalf of the Federation. Any diplomatic training Christopher has is 300 years out of date.
Exactly. Kirk is trained in diplomacy. Part of his job is "seeking out new life and new civilizations". Spock is a scientist. His job is looking into the unknown and finding answers. Christopher is a fighter pilot. His job is to fly an airplane and shoot stuff. How much "diplomatic training" would a USAF officer from the 60s get? He's not even based overseas. What's that line from Full Metal Jacket
? "I wanted to meet interesting and stimulating people of an ancient culture... and kill them."