This doesn't need to be limited to Byzantium, China or western europe in answers. I'd love to hear about why there were Eunuchs in certain cultures and not others and contemporary reactions to them.
You may be interested in u/caffarelli addressing just this lack of Western European court eunuchs. Reactions to eunuchs appears to be ground yet untread on the sub, though; you'll have to wait for a fuller answer on that score.
As a general rule, court eunuchs in places like China, Persia, Byzantium and the Ottoman Empire were employed in two roles:
On top of that, eunuchs existed in and were employed in those places by inertia. Eunuchs had been around sometimes for literally thousands of years and nobody found a need to question that.
The question why Western Europe didn't employ court eunuchs in the same roles is quite simple if we look on the above reasons why court eunuchs existed in the first place. First of all, there was no such tradition to begin with. Court eunuchs would have to be introduced as a novelty and that would cause several problems. Structures of power (like nobles and clergy) were already established and introducing eunuchs would face severe opposition from all concerned entrenched interests.
Secondly, with some individual exceptions (like Louis XV of France) harems were virtually unknown in Western Europe. Surely, kings had several mistresses, but they were as a rule just fleeting romances. Officially kings were good faithful husbands to their wives. Bastards were not eligible to inherit anything, unless some extreme circumstances happened. Just remember that even the almighty Louis XIV had hard time to legitimize some of his own bastards (even born to a woman whom he secretly married as a widower) and even then his will was disregarded after his death.
Thirdly, European kings for most of the time already had an access to a pool of educated men who could perform various administrative roles. Those were clergymen, many of whom (i.e. Catholic ones) were even forbidden to marry. In many respects, Christian clergy filled the roles reserved to Eunuchs in oriental societies. Additionally, bureaucracy in Europe was, up until the Enlightenment era, extremely small. Europe was feudal and most of roles that we today see as naturally being reserved to the state, was in fact performed by private individuals or corporations of people (such as guilds, estates and so on). The number of available bureaucratic positions was therefore extremely limited and the number of clergymen was more than enough to fill them. This obviously changed when Europe entered the time of Enlightenment and later Revolutions. The bureaucracy expanded rapidly. But by then corporal mutilations were already seen as abhorrent and barbaric, therefore it would be absolutely unthinkable to castrate men to employ them as bureaucrats.