This is is very thorough and well argued article from a brilliant young scholar, Gordon Dakota Arnold. He sympathizes with the perspective of more accommodation of traditional, conservative Christianity in public life. The article is a good reminder that America's Founders weren't always on the same page. But we can make observations like James Madison and Thomas Jefferson had a particular vision of church-state relations that was more secular and "separation of church and state" oriented. This has been called the "Virginia view" because Madison and Jefferson were both from Virginia and saw their vision validated in the Virginia Statute for Religious Freedom.
But there were other perspectives; the "Massachusetts view" was most notably articulated by George Washington and John Adams and permitted more expression of religion in public life and more interplay between church and state.
But onto Arnold's article. A taste:
Was Madison a Christian?
Madison and the Great Divorce of Christianity and Politics
Because he believed that religion is essentially a passion that causes rather than discourages faction, Madison also contended that it needed to be pacified for liberty to be preserved. The primary method of solving the political problem of Christianity was to encourage religious diversity and foster disunity. As Madison’s friend, neighbor, and first biographer William Cabell Rives reported, the President was fond of quoting Voltaire’s maxim that “if one religion only were allowed in England, the government would possibly be arbitrary; if there were but two, the people would cut each other’s throats; but, as there are such a multitude, they all live happy and in peace.” And Madison himself left no doubt that these were exactly his sentiments. He spoke in Federalist no. 51 of how the “multiplicity of sects” was the only security for the preservation of “religious rights.” In a letter to Thomas Jefferson, Madison celebrated the fact that the “mutual hatred” of Virginia’s Christian denominations “has been much inflamed.” He added: “I am far from being sorry for it, as a coalition between them could alone endanger our religious rights.” Where the Apostle Paul spoke of the need for harmony, unity, and love within the body of Christ, Madison preferred that the church be characterized by disarray, discord, and faction. Only then would Christianity fail to mobilize itself as a political force, and only then would the natural rights of individuals be safe from a majority faction. Madison’s view, too, contrasts with the more Pauline beliefs of George Washington, who celebrated the “harmony and Brotherly Love which characterizes the clergy of different denominations” because it further substantiated his conviction that “Religion and Morality are the essential pillars of civil society.”