You left out an important piece of source material, notably
Article VI of the US Constitution, with says, in part:
The Senators and Representatives before mentioned, and the members of the several state legislatures, and all executive and judicial officers, both of the United States and of the several states, shall be bound by oath or affirmation, to support this Constitution; but no religious test shall ever be required as a qualification to any office or public trust under the United States.
Sorry, replace "with" with "which".