Naculan Royal Court

At the very least, George Kirk was killed there (the Kirk parents hadn't actually been named at this point, but I'll use George and Winona for convenience). Trivers's plan envisioned George's profession as rather like his son Sam's, a sort of Starfleet researcher leading a team of other scientists on the colony. Their work seems to have involved a long-term project, so it makes sense that Kirk (senior) would have brought his young son with him; in this version, the Kirks had already been living on Tarsus IV for five years when Kodos came to power.

(The timeline established in "Shore Leave," produced very close in time to "The Conscience of the King," would confirm Kirk's youth from this version. When the Kirks arrived on Tarsus IV in this version, 25 years before the main events of the planned episode, Kirk could have been no more than 9 given the information in "Shore Leave," and was more likely 7 or 8. S2 later establishes that he would have been 8, with TOS!Sam Kirk envisioned as ten years older, so 18 and off on his own career. Five years later, during the final genocide, James Kirk would have only been 13.)

Anyway, in this version, there was a lot of upheaval that led to Kodos coming to power in the first place, and George's opposition to Kodos's faction got him killed. We don't know if Kirk's mother was involved with this or not, or even present or not, but if she was, there's no reason to assume she was magically spared in the long run, either.

If anything, the implication that George would have lived if he'd been willing to play nice, and his son surviving that incident only to later be found among the eyewitnesses to the execution speech, could suggest that it may have been Winona rather than George who was considered genetically inferior by Kodos's eugenics rationales. If so, she would have been among those executed, still leaving Kirk as the only eyewitness from his family.

(There's an idea I've seen floating around, I'm guessing derived from the AOS movies, that Kirk's "genetic inferiority" is due to some idiosyncratic disability rather than anything to do with his family. There's no hint of him having that kind of disability in TOS, though, which repeatedly confirms that he enjoys perfect physical health apart from tension headaches and a bit of yo-yo dieting. Even the idea that he has bad allergies seems pretty unlikely in TOS, to go by his repeated interactions with flowers.)

In any case, the idea of Kodos escalating from an already-brutal rise to power to full-on genocide, with casualties like George Kirk along the way, was removed by the demands of the production staff rather than Trivers's own vision. The higher-ups didn't want so many family details nailed down so early, restricting what TOS could do with Kirk's parents later. Trivers ended up coming up with an alternate scenario in which Kirk would have to be older than Shatner really was at the time (rather than slightly younger), removing the whole question of Kirk's parents by having him be a young midshipman selected for survival during the genocide. But that backstory was also removed before the episode was filmed, iirc, and certainly isn't in the aired version (I don't know if any reason for this removal has been given; it's internally contradictory in multiple ways including the basic timeline, though, so not surprising).

And then in the end, TOS never did do anything with Kirk's parents. There's no indication in the show as to whether they're alive or dead, and we're told nothing about them personally, either, so Trivers's original, simpler concept never did get contradicted by anything in TOS.

Q2— Kirk and Tom Leighton are friendly enough that it's not odd for Leighton to invite him to go see a play, and Leighton's wife Martha is obviously familiar with Kirk and on good terms with him. However, Kirk doesn't remember much of anything about the other seven survivors. Leighton and Kirk have pretty accurate ideas of how many other eyewitnesses there were, so it doesn't seem like they never knew about the others; Kirk just doesn't have any personal relationship with the other ones or remember their names. There seems to be some special tie between him and Leighton specifically.

A: In Trivers's concept, Tom Leighton was George Kirk's research assistant on Tarsus IV. By the time of the genocide, he'd already been a familiar part of James Kirk's life for years.

Q3— Kodos's genocide speech, perfectly remembered by Kirk, begins with the statement, "The revolution is successful." But the official historical record, reported by Spock, doesn't mention anything like a revolution, except Kodos declaring emergency martial law as governor, the same summation the computers recited to Kirk earlier. The executions are described by the official record in highly mechanistic terms, as something Kodos ordered, quick and painless but carried out on a mass scale, without any hint of combat or warfare. Kodos's reference to a revolution could be a self-glorifying political metaphor or something, but there are other hints that something bloodier and much more violent happened prior to that speech.

Riley frantically insists that he personally saw Kodos murder his father and his mother. The argument between both survivors as Kirk talks Riley down is distantly overheard by Kodos, and the familiarity of one of their voices triggers his own memories of what he did as "a soldier in a cause" on Tarsus IV. Tom Leighton says he remembers "that bloody thing he [Kodos] did" as the camera focuses on the mask concealing the maimed half of his face. That doesn't seem at all likely to be the result of only the sterile, quick, painless deaths of the official record—for one, Leighton wasn't killed, and for another, the implied mutilation he experienced certainly wouldn't have been painless.

A: Kodos wasn't the purely cold, civilian authority ordering distant executions. His faction's rise to power was violent and dangerous in the first place, and his opponents were brutally punished quite apart from the famine.

For the people of Tarsus IV, the escalation to genocide in the wake of starvation didn't come out of nowhere. The mass executions were quick and painless, but they were the culmination of Kodos's policies, not the beginning nor the only thing he did. Y'all, he's a space Nazi, it's not subtle.

And in all probability, the main reason that so few survivors had ever seen or heard Kodos is that most people who personally interacted with him had already been killed or were among those executed. Leighton may have escaped with his face mutilated, but most others—like George Kirk—never escaped at all.

There were some other details that I thought the basic concept explained much more easily than the various "well, actually, in the lore" gymnastics, too, but this is already long enough. The point is more ... I think "The Conscience of the King" tells us what we need to know for the power of the episode to work—the exploration of the legacy of genocide, all the more when it was aired; the fantastic characterization and exploration of how people engage with these kinds of atrocities; the dehumanization deployed subtly in Kodos's manner of speech and more bluntly in Lenore's—but even when we account for everything in TOS, we're also given plenty of room to explore and imagine and interpret for ourselves.