Okay so I have a lot of thoughts about the whole thing of the Gerudo being a race of entirely women, with the only exception being one man born every hundred years, and that man automatically being their king. Now this worldbuilding comes from Ocarina of Time, and there's obviously a metric fuckton of unfortunate implications there, because it was 1998. And it seems that Tears of the Kingdom is sticking with the lore of Gerudo men being extremely rare and becoming the King of their people, which once again has a metric fuckton of unfortunate implications because it's 2023 and Nintendo has somehow gotten even worse about this shit.
But let's set aside the whole... everything, and look at this from just the in-universe perspective. How does it work? I mean, it's pretty clear that there is no overlap between the kings; the old ones are normally long gone by the time a new one is born, but the Gerudo manage to take care of themselves during the hangtime. So they must have an established system of government and leadership that doesn't involve a king, and somehow that system is set up in a way that does a smooth transfer of power once a new king is born and old enough to take the throne. But why bother always declaring a random guy to be your King when you already have a perfectly functional system in place?
I mean again, the whole thing has a lot of sexist implications, but we're not looking at this from a real world context, we're examining it in-universe. And we could just go the lazy route and say that their king is in charge just because he's the only man, but I don't like that. I mean come on, the Gerudo are a race of entirely women, and most of their outside problems come from Hylian men being creepy about it. They are entirely a matriarchy; there is literally no reason for their culture to have an inherent respect for men, even if the man in question is one of them. And they're desert people; they live in an extremely harsh and dangerous landscape, if they don't have their shit together, they will die. By sheer necessity, their culture needs to put a lot of value in being practical, because if they're stupid about things, people die. They really can't afford to have a shitty leader take over, and just letting some guy take the wheel doesn't really fit with the way their culture must otherwise work.
So again, why the fuck do they bother having a King?
I think it's mainly just a ceremonial position. Yes, if the guy is a good leader he'll be in charge, but if he isn't good at being a King or isn't interested in the job... fuck it, they've already got a functional government system that's been leading their people the whole time, why fix what isn't broken? The title of Gerudo King isn't about leadership or power. I think it's more about belonging. Because the Gerudo are a culture where every single one of them can be defined in the same way... and there is exactly one exception once a century. Men are considered to be inherently outsiders at the best of times, and more often they're enemies. A man born into this culture is a natural outsider; he is completely unique, and that means he doesn't really fit into his community. And well... when someone is fundamentally different from the rest of their community, they tend to be ostracized.
So I think that's why the position of Gerudo King exists. It isn't about them needing or even wanting a man to lead them. The title of King doesn't need to involve any leadership at all. It's about giving the man born every century a place in their society. It's a way of saying yes, you are one of us, you are a Gerudo, you belong here, you are wanted and you are loved.
The Gerudo know that every hundred years, one of their children will be fundamentally different from all of his peers. And so their society is built to ensure that a child who is completely different from them will still be loved and accepted. He will always have a place in their society. He doesn't need to earn their love, he has it just for existing. These are his people.
The title of Gerudo King isn't an inherent position of authority. It's a promise of acceptance.