While the idea of American radicalism ever being very influential is intriguing, the reality is that from their founding, the colonies en masse were quite steadfastly religious and after the Great Awakening in the 1730's - 40's, even more so.
(Yes, acknowledging that there was an Enlightenment era Deism mildly popular among some of the colonial ruling elites).
And keep in mind, the French experience was far more anti-clerical than anti-faith. The Anglican church, the closest parallel, didn't have anywhere near the role that the Gallic Church played in France.
So, the prospect of an anti-religion element taking control during the rebellion - well, that gets into the high fantasy side of alternate history stories.