All of the history I have read clearly indicates the the American Experiment is the product of the Magna Carta (Challenging the authority of divinely appointed Kings by Rome) and the Enlightenment (Voltaire, Locke and Newton).
What a bizarre claim to that which the Church fought for centuries.
All of the history you have read was written by WASPs, none of whom had any interest in relating the foundational documents expounded in the original post.
The Magna Carta was written by Catholics, to Catholics, for Catholics. No other Christian faiths existed at the time of the Magna Carta, so I fail to see your point.
Also, I hate to break the news, but the majority of the founders were affiliated with the Church of England. Most colonists were of the English persecution and many did not come to the colonies to avoid religious persecution (Hence, their continued allegiance to the dogma of the Church of England).