Why the church agree to separate from the state?
For the U.S., the edict that Congress would make no law
establishing a [state] religion, nor prohibiting the free exercise
of religion, was written into the Bill of Rights. Many people had
emigrated here to avoid persecution at home for their religious
practices — especially not accepting the beliefs of the
Church of England — and they wanted to be certain that
that freedom continued. Examples would be the Puritans, Quakers and
others who came later. For the USSR, their Constitution explicitly
stated that the church would be separate from the state, and the
school system from the church. There's no indication that any
churches agreed to this or not; it was forced upon them.