Eunuchs were used well into the Byzantine period and later, but they weren’t particularly numerous. They may have played a role in corruption, but the issue is really dealing with homosexuality and immorality, which does have a marked influence on a society on all levels and was remarked upon by contemporaries in Periclean Athens, for example, and by St. Augustine in his Confessions.
You’ll also notice Nixon’s point just after Archbishop Sheen’s. They’re all making the same argument for immorality, and homosexuality, as the source of the decline. of course, there’s also Sodom and Gomorrah...
At the same time, though, I can’t but think that homosexuals were more a symptom than a cause.
The best evidence of this is that at its height, there would have been at least some homogeneity (no pun, really), throughout the major centers of the Roman Empire. But it was a thousand years after the collapse of the western empire that the eastern, Byzantine empire folded.
Far more geographic oriented, with the western empire under pressure from the barbarians.