This is a rather heavy question you've put up here... And I get the feeling you're not looking for a nuanced 3-volume exploration of cross-cultural liminal gender theories, so I'll see what I can do you in three paragraphs, if for the mental challenge if nothing else! If I've got the wrong read on you though, let me know, there's a few places I can point you if you want to make this your life's calling as well. Also put on your thinking cap because you are about to get on Mr Bone's Wild Gender Studies Ride. Handily for me you have skipped the Italian castrati who are kinda outside of some of these anthropological theories, but also aren't, and I'm a bit too sleepy to deal with that this morning.

But the simplest answer is: these societies made eunuchs because they wanted to have eunuchs. And what does "eunuch" mean in a society? In these societies, eunuchs were a highly specific class of people: you can frame them in whatever way you'd like, as their own race of people, as their own gender of people, however you'd like to think of them, just that in these societies there was a call for an Other to fill certain social roles, so they made one. And eunuchs were an Other so very other that he is outside of a binary gender paradigm, with his otherness inscribed on his flesh through the act of castration. This creates a permanently liminal (threshold, or in-between) gender, which has many uses in a society with high levels of segregation of some sort (man-woman, man-god, man-royalty, etc.) Now, do you NEED to castrate someone to make them a social Other? Of course not, and many societies had liminal genders without any physical fuss about it. But these societies didn't. Castration is perhaps useful to think of only as a ritual that created the status of eunuch, because people often get a bit too focused on the physicality of it, and think of them simply as men who just happen to be without their sex organs (which is how we mostly think of the castrated today in our society as we have just about completely lost the "race" of eunuchs), but they were not injured men in those societies, they were eunuchs.

So you've got your Other now, where does he go, what does he do? He goes to the gates, the locked doors, the sacred spaces, the forbidden and secret, he goes wherever you need a go-between. In your litany of societies above, the eunuch is a creature of the court, an elite servant, a politician, a religious servant of God, a teacher of children, a link between sequestered women and the outside world, or perhaps just a house-slave that mans the front door. His roles gravitate towards other liminal venues in society, be they keeping gender segregation intact, protecting children from adults, keeping the common people away from the royalty, keeping mortals a respectful distance from God, or just keeping unwelcome people out of your vestibule (unexciting, yet somehow the obvious liminal space here!)

Most people who aren't into anthropology or gender studies will give you the same cheap answer - harems. In fact both of the 2 deleted answers here are "because harems and sexual access." I suspected they would be when I saw the IFTTT hit for this question in my inbox this morning, this idea is so prevalent! Anyway, you can say "because harems" sure, but it only explains part of what they did in any society that had them. That's the cheap answer. This is the bespoke haute couture answer. As such, it is perhaps not for everyone, but I hope you get a pleasant bit of brain-stretch out of it anyway. :)