The religious parts of this I'm not sure I can agree with really.... I'm orthodox if it matters or seems relevant.
The catholic/orthodox differences are largely just because of a thousand years of divergence, of speaking different languages and having relatively little interchange, of each individually having movements in response to internal pressures and trends not experienced by the other. Not going to get into a filioque debate on HN but the initial theological dispute, however significant you find it to be, is not the source of the most tangible differences in the two branches today. They've just each been doing their own thing for a millennium and their unique histories took them to two different places during that time.
I don't see how byzantine eunuchs indicates anything about priest celibacy, especially since orthodox priests are usually married. Eunuchs and celibate priests still come from families, they experience love and duty and allegiance and enmity. To the extent a position is admirable people will want to be in it and to the extent it's powerful people will use that power to benefit the people and things they value. No restriction on who can hold an office will by itself address those factors. Byzantine eunuchs got up to plenty of corruption and betrayal in their own right.
I assume by "barbarians" you mean the ottoman turks, but we have to be careful in reading byzantine history not to absorb byzantine attitudes about their rivals. The ottomans were a long-lived, sophisticated, and nuanced entity in their own right. Even their precursors and other byzantine neighbors were not as simple or simply motivated as byzantine or byz-sympathetic sources would indicate.