Interesting point. The Compact certainly reflects the deep Christian devotion of the Pilgrims. The whole world was a much more religious place then and, if you read just about any official document or proclamation, you would find it replete with such dedications to G-d, etc.
As you correctly point out, the Pilgrims came to escape religious persecution, as did many of the early settlers who crossed the ocean. When that motive found its way into the founding consensus of the new country in 1776 and later, they clearly intended to address religious persecution by making sure that no one religion should rule. I still agree with my Christian friends that the great founding concepts and sense of public morality stemmed very clearly from the unmistakeable (Judeo)Christian traditions of the founders. Therefore, I am in wholehearted agreement that America was, and probably still is, a Christian nation. Unfortunately, the last 50 to 60 years of liberalism has stripped almost all of the best part of those Christian roots out of the national discourse and the cultural life of the country.
To anyone who thinks it is a paradox for an Orthodox Jew like myself to rejoice in the Christian origins of America, it is not because I make any compromise in my own faith. It is simply because the founders gave the nation all the best attributes of Christianity that can be celebrated by men of any creed: charity, loyalty, love of peace, good will, fairness, natural law, liberty and even a healthy dose of hard work and self-reliance.
Fulton Oursler began writing the The Greatest Story Ever Told in 1935 and concurs with your dating.
It was Rabbi Solomon B Freehof, of a great Jewish temple in Pittsburgh, who said to me at dinner one evening that the unspoken scandal of our times was the hidden fact that Bible-reading had been largely given up in America.