The earliest stardates on the series started with a 1, while by the end of the show, three years later, they started with a 5.
Umm, why do you think it would be three years later? It was a five-year mission, after all!
It cannot be three years, that much is certain: in the early third-season "Day of the Dove" (SD unknown), the interval between the episode and the late-second-season "Errand of Mercy" (SD 3198) is given as three years by the character Kang. It's at least four years in three seasons, then - and the stardates would nicely suggest it's actually five years and we saw the entire adventure.
As for consistency, following the rule that higher stardates establish later date creates lots of it. Say, Chekov is definitely onboard in "Space Seed" already, and thus may be among the people who flocked to see, touch and hopefully have sex with Khan. Saying that a thousand of the units amount to a year creates further consistency regarding the time intervals discussed. And it's even within tolerances to think that we witness the resulting roughly three stardates per day...
It's pure coincidence, of course. But so what?