The results are, that for 60 or so years after the Constitution and Bill of Rights, people in power, meaning Presidents, Congressmen, judges, etc on occasion did things that were incongruent with the Constitution and Bill of Rights. What is really amazing is that such occurrences were, in truth, so rare. The trend, even during the early 19th century, when in many respects we did have a de facto unconstitutional established Protestant version of Christianity, was still more and more towards the total separation of church and state that was written into our Constitution and Bill of Rights. The process was begun in 1789, grew slowly at times during the early 19th century, picked up speed in the latter part of that century, and really began to speed along in this century till we are where we are today. In a perfect world there would be no incongruencies, but we don't live in a perfect world.
Men are human and sometimes do things that give mixed signals.
The total picture, even with the few mixed signals there are, clearly indicates a complete separation of church and state was needed and intended to have true religious freedom. No it isn't always easy to find that line of separation, but that line does exist. Chaplains in Congress, paid for by tax dollars, as well as proclamations of thanksgiving do sit on that line, and at times cross that line.