Meacham argues that while the Founders deliberately created a secular state they also created a "public religion". American public religion is a sort of generic religion without any creed or troublesome details. It reveres a creator God who is just but merciful and who guides events with a vaguely defined Providence. It allows for public piety without establishing a particular sect as the official state religion. Public religion seems a lot like Protestant Christianity with the Christ part left out, and in every generation forces of organized religion have tried to drop the pretense and declare it officially Christian, but the Founders carefully worded the Constitution in such a way as to keep that from happening.
Faith has played a huge role in American history, both positive and negative, and we cannot wish it away in the name of pluralism and separation of church and state. The genius of the Founders was that they found a way to govern a diverse, quarrelsome, and highly sectarian nation by publicly embracing religion but doing it in such a vague non-specific way that they did not offend anybody.