The big thing with being a eunuch is that the idea was that because you gave up having direct descendents, it meant that you could more easily be trusted to run bureaucratic tasks for the good of the royal line. Whether or not this works is one of those oft-debated ancient and medieval political science topics of the time (particularly in China) with most petty lords that would lead revolts against kings/emperors blaming the bad government on the influence of eunuchs. If this was true or not is unknown, but the ones that won were the victors so their histories survive.

As far as I am aware though, you are correct that the only culture that believed eunuchs to be 'sexless' is the Catholic Church.