I have, elsewhere on this blog, defended the position that America was founded as an unorthodox Christian nation, and also that one sign of that unorthodoxy was the complete lack of bishops of any episcopally organized denomination (Anglican, Catholic, Lutheran, Orthodox) in America prior to the revolution.
A tidbit relating to this, that I have yet to integrate into my view, is the following: having rejected bishops and the ecclesiastical machinery that goes with them, one of the institutions lacking in America was religious courts. Because of this, offenses that were matters for church discipline in England became matters for the civil magistrates and courts in America. Among them Brogan lists (Longman History of the USA, p.44): heresy, witchcraft, profanity, blasphemy, idolatry, adultery, sodomy, and Sabbath-breaking.
As I have mentioned earlier, the cause for Roger Williams' sermons against the involvement of civil authority in church affairs was that the civil machinery was coming down on him and his deviant congregation for their refusal of communion fellowship with the Anglican church (and thereby with the Puritan congregations of the Bay colony), a purely religious matter.
So, rather than separating church and state, American unorthodoxy pulled the state in colonial times into a broad spectrum of affairs that weren't its jurisdiction in contemporary Europe, to fill the vacuum left by the missing church hierarchy.