Unlike most churches in the USA, in the Orthodox tradition, the Church is a National Church. There is a Russian Orthodox Church, a Greek Orthodox Church, a Bulgarian Orthodox Church, and so on.
So in a fundamental way, the church is an arm of the state.
The Church of England established by Henry the Eighth adopted this model, but the US Constitution specifically prohibited any laws establishing a State religion.
Although the Orthodox churches are established by their Governments, they maintain their moral authority, like the conscience of the Government. They function as a moral watchdog of sorts - more in influencing policy behind the scenes, than in confronting the Government’s policies publicly. In fact, Orthodox Churches teach and advocate patriotism and support for the Government as part of their doctrine - that it is a good Christian’s moral/religious duty to be a good citizen of the State.
The Church however, also remains bound by some fundamental aspects of Christian doctrine, which could put them at odds with the Government, if the actions of the Government crosses certain extreme moral or doctrinal red lines - which would be their equivalent of a Constitutional crisis.
Well put. It is no wonder the law prohibiting the establishment of an official state church is enshrined in our Constitution. Right on top of the list, no less - as the 1st Amendment. Our founders were doing people of faith a big favor.