To be contrary, I think they were very worried about this. There were many different pastors of many different denominations who discussed this at all levels. The idea is what started the discussion of natural versus God-given rights; state religion, etc.
None of the Founding Fathers were deficient in their knowledge of the Bible, or their knowledge of the history of Europe. The different discussions going on publicly and privately show the great amount of thought, discussion and prayer which preceded the Declaration and the Constitution.
So you honestly think that if they thought it was a sin that we’d be British subjects today? Insanity.