I am having a brain lapse. When did Enterprise return to Earth during the course of TOS? I mean real, 23rd century Earth. Not Guardian of Forever Earth or black star slingshot 1967 Earth but good ol' 2265 Starfleet Earth?
Well, those two times, obviously. The only rational way to get to this "slingshot Earth" to do historical research there is to go to the "real" 23rd century Earth first and then slingshot. Although clearly you can also cast all rationality aside, slingshot around some black star somewhere (perhaps within the Sol system, though, since the region has its share of such anomalies and has been suspected of harboring at least two lack holes, in ST:TMP and "Past Tense"), and still end up in Earth's atmosphere somehow during the brief time you are unconscious - but when you return from that predicament, you return by slingshotting around Sol, and thus meet the "real" 23rd century Earth anyway.
Kirk basically stayed right next to Earth when visiting Pollux, too. Or Vulcan. Which happened to be almost en route between his previous assignment and the also-next-to-Earth Altair. None of which meets Kirk's own criteria for the final frontier, because he routinely quoted distances in the high hundreds of lightyears and visited stars located even farther our.
I'd say for the most part they were doing what their narrated mission statement said they would do AND doing the mundane stuff like escorting diplomats, fighting wars, policing space lanes, etc.
Yup. It's just that the mundane stuff appears to account for way more than half of what he was doing during those supposed five years... Either he stayed close to home, or interacted with peoples already quite familiar to the UFP, or both.
Of course, there's no harm in mislabeling the product in promotional material, which this "opening speech" would probably represent if it were somehow part of the Trek universe.