Catholics and Deists (Thomas Jefferson, John Adams, and Ben Franklin being central Deist figures) were as present in the founding of the nation, and all the major advances and things that made this country great came during periods of plurality and escape from rigid puritanical values, while the times of retraction came from an attempt to "return" to something less ideal.
I consider it an accident of history which religion was a majority (esp. since, back then, religion was moved out of favor) in the creation of the first modern democracy. If anything, it seems like the greatness of the U.S. come from the strong need to counter puritanical extremes. I guess, in that sense, we need it.