That is precisely what some militant atheists, secularists, and even some religious leaders want to have happen today. These folks are what I call "separationists," those who believe that religiously based moral values ought to have no place in public discourse or policy-making. While most of them merely disapprove of such an interaction of religion and politics, others are so hostile to religion - especially conservative Christianity - that they would formally prohibit it.
Separationists come in different varieties, all of which provide examples of how not to think about the relation of Christianity to the political sphere.
There are the militant atheists - Richard Dawkins, Daniel Dennett, Sam Harris - who find religion so dangerous they seem to want it banished legally from public life.
Others - not so militant - want religious people to drop their religiously based moral values when they enter the political sphere. They want only secular, rational, and purportedly universal values to enter the public square. They consider religion so parochial and irrational that it will likely lead to some form of theocracy if it has its way in the public sphere. For example, they are appalled that religious people have effectively supported policies that limit abortion and wrongly say such actions are a violation of the separation of church and state.
The First Amendment does indeed prohibit the establishment of a specific institutional form of religion (separation of church and state), but it guarantees the free exercise of religion, which historically has led to the lively involvement of Christian individuals and organizations in political life. Separation of church and state is quite a different matter than interaction of religion and politics.
Moreover, limiting Christian activity to the private sphere violates serious Christian belief, which affirms that God is active in all facets of life and that Christians are obligated to follow his will in them. Separationism goes counter to the Constitution, American history, and serious Christian conviction.