I find the American religious roots thing to be largely a farce. Founding American politics was about satisfying a diverse theological group. While language was somewhat religious, the ideas and structures were enlightenment. American government was free from the authority of an official Church-- one which would both raise moral concerns in a formally, manage charitable services and remind state of its limitations. Since then, American politics has paid lip service to religion but has not engaged a particular theology as an equal. Many of the European countries with many more non-religious people maintain a traditional relationship between Church and State.
I am fairly certain history is chalked full of non religious people. In many ways people are religiously non-religious; they look to science or scientists for answers to life's questions. Since science as seen as the "right" way of viewing the world, ethics come from interpreting scientific findings as oppose to interpreting holy writings. If you put "science" as a choice of religious affiliation, how many "nones" would happily take that choice and how many would find it absurd? That is an experiment I would be interested in running.