First, there is no “Judeo-Christian tradition.” Judaism rejects the Christian God, and Christianity claims that the Messiah/Christ fulfilled the Hebrew religion in a way that Jews don’t recognize as valid. No one in the USA at the time of the Revolution thought of them as one tradition.
Second, absent Christianity, there could have been no Enlightenment. Pre-Christian Greco-Roman religion elevated youth, wealth, power, and virility and disdained poverty, age, weakness, and infertility. Christianity more or less flipped that on its head. In that light, we can see how the Enlightenment is essentially an offshoot of Christianity–and understand why some believe that weakening the social influence of Christianity means endangering Enlightenment legacies. It also explains why I think that even the revolutionaries who considered themselves members of the Republic of Letters were actually influenced by Christianity.