Yet the "five year mission" of the Enterprise can't plausibly mean much else than a retrospective highlighting of those five years when Jim Kirk commanded her.
Not likely, since the prologue of each episode plainly stated "These are the voyages of the starship Enterprise. It's five year mission... To boldly go where no man has gone before". While a degree of poetry and hyperbole are permissible given the fact it is a prologue, it nevertheless can't go off half-cock claiming it is about a five year mission to go where no man has gone before if it is about the five years that just so happened to be the captaincy of James Kirk during which he went where people had already been.
What was seen in TOS is certainly not as open and shut as you claim. The fact that only starships like Enterprise were seen and that sometimes they were seen returning to base for resupply and repair can just as easily mean thirteen starships pushing beyond the frontier, zig zagging back when needed to distant starbases, on a five year deep space mission.