I was under the impression that the Connies were all sent on five year missions.
That's a widely held fan assumption, but in actual canon there's not a shred of evidence for it. There's never been any mention that any ship other than the
Enterprise had a five-year mission, ever. And outside of the opening narration, there are no explicit references to the 5YM within TOS itself; the closest we come is in "The Mark of Gideon" where Kirk says the ship has enough provisions to feed a crew of 430 for five years.
The few explicit references are post-TOS. In ST:TMP, Kirk cites "My experience, five years out there dealing with unknowns like this." The only actual use of the phrase "five-year mission" in all of Trek outside of the TOS main titles is in VGR: "Q2," in which Icheb states "Finally, in the year 2270, Kirk completed his historic five-year mission and one of the greatest chapters in Starfleet history came to a close." (Maybe one could make a case for "Yesteryear," in which McCoy says that Thelin has been Kirk's first officer for five years, but that's inconclusive, since they could've served together before the tour of duty began, and it's an alternate timeline anyway.)
If anything, Icheb's line suggests that Kirk's five-year mission was something unusual -- not just
a historic five-year mission, but
his historic five-year mission, as though it were something particular to him. It also implies pretty strongly that the mission of 2266-70 was his only five-year mission.
Hell, the ship was actually 20 years old BEFORE Kirk took command.
We have no firm canonical information about exactly when the
Enterprise was commissioned. The only canonical statement about its age, Morrow's "20 years old" line from TSFS, is clearly incorrect. We know the ship was launched before 2254 ("The Cage"), but beyond that, we only have conjecture. The Okudas' 2245 launch date was pure guesswork.
Why could the Excalibur, Lexington, etc not also have gone out under 5YMs and all completed them before moving on to more core-world patrol and exploratory missions once the Klingons started rattling their Bat'leths.
There's no reason they couldn't have, but there's no evidence that they did, and it's illogical to assume that
every starship mission is five years long by default. One isolated example does not prove a universal pattern. It's possible that multiple ships have had five-year tours of service, but that's far more likely to be just one possible mission profile out of many.