TOS seems to imply only the Constitution ships are "starships"
Naah - the ancient Archon
was considered a starship as well. So it's a continuum of starship classes even back in TOS. And while some of those might not be capable of mounting five-year missions, some others would be: the Constitution
class wouldn't have appeared out of nowhere, without direct predecessors.
The idea of a Starfleet solely consisting of Kirk's ship and any copies thereof TPTB could afford to show is not a particularly defensible one. Star Trek could rarely afford to show, but it made up for that by telling. And if Klingons can muster eight (unseen) ships to face a similar Starfleet force over the relatively insignificant Organia, then total fleet strength isn't something we should count in the dozens, but in the hundreds at the very least.
Kirk being unique among the available officers is really evidence of something exceptional. Whether it's 5yr missions being rare, or Earth being exceptionally devoid of officers at that time (and perhaps others), we don't know exactly. It may still simply be evidence of Kirk lying to Scotty, too...