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 Any diplomatic training Christopher has is 300 years out of date.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."
[From
Errand of Mercy]
AYELBORNE: It is our way of life, Captain.
KIRK: That's the first thing that would be lost! Excuse me, gentlemen. I'm a soldier, not a diplomat. I can only tell you the truth.
[From
The Mark of Gideon]
SPOCK: We must acknowledge once and for all that the purpose of diplomacy is to prolong a crisis.
MCCOY: Well, what are we waiting for, Mister Spock? We're not diplomats.
SPOCK: We are representatives of the Federation, Doctor.
MCCOY: That doesn't mean we have behave like children and listen to some fool lecture by a diplomat!
SPOCK: Unfortunately, diplomacy is the only channel available to us at the moment. The planet is shielded from our sensors, therefore we cannot scan it. Therefore we are unable to select co-ordinates. They must be provided by Gideon.
CHEKOV: We should never have agreed to such restrictions.
SPOCK: We did not, Mister Chekov. The Federation did. Lieutenant Uhura.
UHURA: Yes, Mister Spock.
SPOCK: Contact Starfleet immediately. Explain our problem. Request permission to use every means at our disposal to press the search for Captain Kirk
From these two excerpts, it seems that Kirk and Spock either did not receive as much diplomatic training as you suggest, or they didn't really care too much about it. In both of these cases Kirk and Spock respectively are frustrated by diplomacy and would prefer a more aggressive strategy. Also, USAF Fighter Pilots do more than just shoot stuff, their training (both physical and intellectual) is very demanding and they are required to use their own judgment at times.