I think the big difference here is that while a Christian might say that *insert sin here* might be worthy of death, our religion is to forgive and pray that God will help them before it is too late.
We have no equivalent to sharia. We are not a legalistic faith. Islam is inherently legalistic because it is a political entity.
So no, really, I wouldn't say, "Oh no, those poor innocent fags" is the type of argument we use. They really deserve the punishment, because they are not innocent. We instead would say, "He who is without sin cast the first stone", and an honest person will be forced to admit that in some way they have gotten away with much, and have been forgiven much. Passing it forward is the right thing to do.
And truly, secular government in the west became a thing because of all the religious wars that protestants would fight. This was the way to keep the peace.
Honestly, secular non-religious government that is distinct from the church is probably the most Christian form of government. A government that respects freedom of belief.
The type of secular government that has an aversion to religion would not be included in this. Seperation of church and state in the sense that the government cannot display anything resembling religion is a dangerous idea perpetuated by communists who know damn well that if the state owns everything this form of secularism will be used to eradicate religion.
The Orthodox Church is certainly a hierarchical church, and that sort of thing is also obnoxious to communists, who hold all forms of hierarchy with suspicion.