I want to step out of my Puritan specialty for a moment to record a passage from an old American history textbook that speaks to the problems of connecting the first 17th-century English founders and their religion to the Founding generation in the late 18th century.
I was reading the Popular School History of the United States, part of Anderson's Historical Series, published in 1889. On page 67, the author pauses to fill in the outline of the Pilgrims:
"Our fathers were brought hither by their high veneration for the Christian religion. They sought to incorporate its principles with the elements of their society, and to diffuse its influence through all their institutions, civil, political, or literary. Let us cherish these sentiments, and extend this influence still more widely; in the full conviction that that is the happiest society which partakes in the highest degree of the mild and peaceable spirit of Christianity."
As modern historians, we would never write in this style. But many people today make the same point: since the first English settlers came here for religious reasons, we were founded as a religious/Christian nation.
This elides the 150 years between the only English settlers who came for religious reasons (Pilgrims and Puritans, not Virginians) and the Founders. All that time of tremendous change and diminishment of the religious angle of settlement is simply folded up, tesseract-style, to make a quick, short step from Plymouth to Independence Hall.
It also ignores the fact that religion was important, but not the only focus of settlement for Puritans and Pilgrims. The Puritans especially intended to make money and live well, and be very successful businessmen.
While the Founders were influenced by their New England ancestors, it's clear that many of them worked hard to keep religion from being the foundation stone of the new nation. Yet it's very tempting to see the Pilgrims in direct, immediate connection to the Founders. While Anderson's Historical Series goes obviously too far, its inheritors can come close to making the same point, in much less flowery language.