Well there are certainly some glaring problems with this episode.
"Buying" an entire planet? I'm sorry... but from WHO? So we have to imagine that there is some species heavily populating an entire solar system, enough to claim ownership of all the planets within it. And then a perfectly habitable planet that they haven't bothered colonizing yet could be sold to someone for the "right price"? It's laughable.
OK, maybe it was an inhospitable planet that Flint could have partially terraformed. But a planet is still a vast amount of real estate, insanely so for just one individual. And the fact that one person would own it and live on it would be something terribly difficult to hide from the media. "Man buys planet -- chooses to live there in solitude!" And yeah, you'd think that the Enterprise would have more knowledge about the planet, especially if it could be looked up in their computer databanks. Perhaps for the benefit of the doubt, the Enterprise transmitted a data request to Starfleet and by the time they got to the planet the data hadn't completely downloaded.
KIRK: I'm Captain James Kirk.
FLINT: I know who you are. I have monitored your ship since it entered this system.
KIRK: Then if you know who we are, you know why we're here... Mister?
FLINT: Flint. You will leave my planet.
SPOCK: Did you say your planet, sir?
FLINT: My retreat from the unpleasantness of life on Earth, and the company of people.
--> Here's my revision:
KIRK: Mister Flint, we sincerely apologize for our trespassing. It was not our intention to do so, as our database records were not updated on this planet. I have a very sick crew up there. We need ryetalyn for the remedy and we can't possibly reach another planet in time.
FLINT: You're trespassing, Captain.
KIRK: We're in need! We'll pay for it, work for it, trade for it.
FLINT: You have nothing I want.
KIRK: But you have the ryetalyn that we need! I implore you, as one civil human being to another. You have my word that once we've secured the ryetalyn we will depart promptly and leave you to your solitude.
FLINT: If you do not leave voluntarily, I have the power to force you to leave or kill you where you stand.
KIRK: Kirk to Enterprise. Mister Scott, lock phasers onto our co-ordinates.
SCOTT [OC]: Aye, Captain, all phasers locked on.
KIRK: Mister Flint, we take threats seriously. If anything happens to us, four deaths and then my crew comes down and takes that ryetalyn.
FLINT: An interesting test of power. Your enormous forces against mine. Who would win?
SPOCK: Mister Flint, unless you are certain, I would suggest you refrain from a most useless experiment.
KIRK: We need only a few hours, that's all
MCCOY: Have you ever seen a victim of Rigelian fever, sir? They die in one day. The effects are like bubonic plague.
FLINT: Constantinople, summer 1334. It marched through the streets, the sewers. It left the city by ox cart, by sea, to kill half of Europe. The rats, rustling and squealing in the night as they, too, died. The rats.
SPOCK: Are you a student of history, sir?
FLINT: I am. The Enterprise, a plague ship. You have two hours, at the end of which time you will leave.
KIRK: With all due gratitude.
So, Kirk didn't have to say outright "if necessary, we'll take it
." There's a more obvious imbalance of position, with Kirk trying to appeal to Flint's humanity, and Flint displaying an uncaring disregard for human life. It's Flint's recollection of the horrible suffering of disease that manages to surface it, so that he acquiesces.