Perhaps the villain just feels that he/she is a superior type and ought to rule the world and run things the way he/she wants.
Also, there are two factors that could allow such a plot to work.
Other than the Gang of Five and Chomper, there appears to be no Sharptooth/Flattooth friendships. alliances. Sharpteeth and Flatteeth aren't likely to help each other unless someone from both groups starts the process of working together to stop the villain.
Other than the Great Valley, I think all Flatteeth are segregated. They'd look after their owns kinds and it may be that they'd not think to work with each other or by the time they did, things would be too out of hand to stop it by then.
Also, another point would be to get a few Sharpteeth and Leaf Eaters to back you. In short, traitors to their own kinds who would be merciless enough to watch their own kinds die or hunger or be enslaved as long as they themselves got to hold power.
I imagine being able to push others around might be as tempting to dinos as it is to other kinds.
As for the what next part, how often do we see that in parts where villains plant o dominate the world or larger spacial entity?
I mean, even in Star Wars, after Sidious turned the Republic into the Empire and massacred most of the Jedi, he and Vader seemed content to just rule with an iron fist. As far as we know, the Galactic Empire didn't invade other galaxies or anything to expand their empire. In fact, I can think of only two places that they either took over directly on screen or indirectly, Tatooine (it wasn't part of the Republic in PT I thought. I know the Hutt's gave them air space after his son got rescued, but how did the Empire come to control the entire planet without the Hutts having a say in that?????) and Bespin, which actually seemed to be free of Imperial rule almost sneakily until Vader came, and he was after Leia and Han and Luke, and his primary motive of going to Bespin was to get them, not to punish Lando.
Also, unlike in Star Wars, obviously in LBT, the bad guys ultimately AREN'T going to succeed (imagine if they did, it would ruin LBT!!!!) and will fail, hence we never find out.
Another case of where the 'what next" is even stranger is in Avatar the Last Airbender (the first one, not the Korrah one) where Fire Lord Soszan notices the Earth Kingdom is often at war a lot and comments that they should invade them to make them peaceful too and help make the world better (in short, we're peaceful, they're not, so let's start a big war to make things better.

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Later, they wipe out the whole Air Tribe, pretty much, just to get at the Avatar (what was the point of that again????

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And, though Fire Lord Oazi says he wants to be the Phoenix King, we never get any hint of would the Fire Nation have planned to do with the world had they won. It appears that ruling it was enough.
In fact, this line of thinking for the Leaf Control plot would be quite similar to the Fire Nation's world domination plot. Just take over as we're obviously meant to rule and who gives a Threehorn's toot about how many die in our glorious scheme. And they had traitors of other nations there too (i..e the Dilee of Bossingsay.) that were helping them at one point too. In the Last Airbender, the Dilee knew they were in trouble so they betrayed the Earth Kingdom in order to be the puppets of the Fire Nation and continue holding power (and not end up in the brig or worse).
Plus, everyone complains about a lack of a really sinister villain in LBT. This type of villain would be so dangerous, especially if he poses himself as a wise and kind dino and lures dinos to give him power and is just very sneaky.
Also, Last Airbender was a hit, had quite a few dark scenes, even genoide, yet was aimed at kids. And it was so popular.
Explain why LBT can't take that angle for a change?
Also, suppose Rinkus and Sierra HAD gotten that stone to work, do you think that maybe, just maybe, they might have decided to take over more than just the Great Valley with just their special powers? I mean, a power grab plot has already happened in LBT. It's not foreign to it.
(Note: I think Pterano would have been content to just control the Great Valley, Rinkus and Sierra I think had bigger plans.)
However, if you must know, I actually DO have such a scenario where the villain thought that he could lead the world through all the turmoil the LBT world is always having. He even thought that, if say, the Sharpteeth and Flatteeth were to die out in a choatic sitatuon due to lack of food, or be declining rapidly, rahter than have to fight them for it then, they'd already control the goods and could ensure the survival of his own dietary type (Bothtooth) and not have to have a big fight to get it in such a chaotic circumstance.
As for his minions, perhaps he planned to teach his Sharptooth minions to eat plants and his Flattooth minioins to eat meat, and, over time, and through genetic mutations, would produce omnivore types from them too. (Who knows?)
Everyone knows omnivore are the most adaptable, and maybe it will go to their heads. I mean, maybe they think, since we have the biggest advantage on survival, we should rule and we can lead the other kinds to be like us, but they aren't gonna just let us take over, so we gotta break their wills.