We have to wonder whether Starfleet sent our heroes to Organia in order to further Starfleet strategic interests (i.e. keep Klingons out at any cost) or to protect the Organians. There are bits of evidence for either of these being the true motivation...
Kirk's orders do mention the strategic importance of the planet, in the sense of it being useable by both sides. But evidently only the Klingons are ever going to actually try and exploit it, so Kirk isn't expected to secure a key asset for the Federation - he's merely sent to harass the enemy on a side theater of the war. This is supported by Starfleet only sending a lone cruiser when entire fleets are on the move.
Kirk's mission really seems to be one of guerrilla action against inevitable Klingon occupation from the get-go: he never appears to think that his starship should have a realistic chance of keeping the Klingons away. He could accomplish that
mission all on his own, that is, with Spock, conducting more of those rather effective two-man raids. So when he starts to incite planetwide rebellion, it seems to be for reasons outside his actual mission orders.
This is classic Kirk: even in the middle of a war, he is mindful of his ideals, which he wants to be everybody's ideals. Native peoples need to be free, regardless of whether they want it or not - here, in "The Apple", in "Return of the Archons", in "A Taste of Armageddon"... Kirk's buttons are easily pushed, and he really does believe in the liberty-or-death thing.
It's not Kirk the soldier, the ruthless servant of UFP strategic goals in action here, then. It's Kirk the uncompromising idealist, the fanatic libertarian, who has no choice other than to send all Organians to their deaths in a hopeless struggle. And all the more so when they see no point in such a pursuit.