I think you sorely misunderstand the Byzantine ecclesiastical structure if you think weakness is what kept Patriarchs and Bishops from going against the Emperor (the Roman/Byzantine one I mean). There simply was no need, the Emperor was Orthodox just like the Bishops were. He lead the people in their material lives while the Church guided their spiritual ones. It was an equal alliance. When one side over stepped its bounds the other acted as a corrective. This can be seen in the iconoclasm controversy, and (though in a way which you will disagree with) the political efforts at reunion during the later middle ages. Just because the Patriarch didn’t want to declare himself a prince that has political temporal authority over all the Earth (something claimed by the medieval papacy) does not make them weak or poor witnesses to Christianity. If anything the Byzantines saw the late medieval papacy in all its temporal “splendor” and decided that it went against the Christian spirit. The medieval papacy’s lust for universal temporal power surely goes against Christ’s teaching where He said
“But Jesus calling them, saith to them: You know that they who seem to rule over the Gentiles, lord it over them: and their princes have power over them. 43 But it is not so among you: but whosoever will be greater, shall be your minister. 44 And whosoever will be first among you, shall be the servant of all.” Mark 10 [42-44]
Aside from this, if you want an example of an Eastern Bishop attempting to rule as a temporal leader, you have only to look to Patriarch Nikon of the Russian Church. Then you can understand why the Russian government reacted so harshly and subjected the Church totally to the state.