It's the Mirror Universe, pretty much everyone in any power whatsoever gets assassinated very quickly after getting it.
In a novella, Empress Sato had
a pair of clones; one of those clones went and killed the other clone off, but clones nonetheless. So it’s not like the Sato Dynasty was dying off anytime soon, and theoretically could have continued forever through cloning well into the 23rd century and beyond. Not to mention all the doppelgangers running around as decoy to protect Empress Sato from assassination.
Even without all that, Empress Sato likely concentrated power as soon as possible, since she brought them the most powerful ship in the fleet to stop the ongoing rebellion and overthrow the emperor. With the latter, which from what we can tell from IAMD, was popular sentiment, since no one tried to stop Mirror Archer at all. Even Sato, after offing Mirror Archer with the assistance of Mirror Mayweather, continued with the plan to depose the emperor.
There's no evidence that anyone who get into power quick in the Terran gets offed just as quick. If anything, it's comparative to the Klingon Empire. And it's known that Chancellors can serve terms that last decades there (ex. K'mpec, Martok). So it very plausible that Sato was still in power when Georgiou won the Hunger Games tournament. We know Sato was ruthless enough to take power; there's no reason to think that she was not ruthless enough to keep it for over half a century.
Also, think about what the movie is selling you about the Terran Empire before Georgiou.
Empress Sato made the Terran Empire somehow a bit more humane than when she found it. Since, from what we can tell from Section 31 movie, power in not concentrated in her family/house and succession is not hereditary, but instead is democratized enough for others to have a chance at power through a tournament. Killing one another and/or superiors while serving of a starship to ascend up the ranks does not seem necessary anymore (and probably good for the military too, as they no longer lose their best military strategists on a whim). And Emperor Georgiou made it worse, reverting everything back to the way it used to be a century prior, including the threat of coups being commonplace, and a rebellion against the empire.
It also means that Empress Sato was a more competent reformer than Mirror Spock, since the Terran Empire did not collapse and be overrun by its enemies under her reforms.
Meaning Empress Sato was the best thing to happen to the Terran Empire, and it was all downhill from there after her death.
And there was definitely some turnover because Sato declared herself Empress but by the time of Georgiou "Emperor" is a unisex term.
Maybe no one is allowed to claim the term Empress out of respect of Empress Sato.