I'm less confident about the other points, but on this one you're quite demonstrably wrong.
(1) The standard at the time was to declare a state religion, but the United States did not. If the establishment of religious intolerance was a goal, they could easily have done so.
(2) There was a considerable push by the religiously intolerant to get a stronger footing in the Constitution, which was rebuffed. There are no references to theism in the Constitution or Bill of Rights, save that, as was the custom of the time, the year was labeled the "Year of our Lord one thousand etc."
(3) Several of the Founders were Deists or Unitarians, who are about as tolerant as you can get while being religious at all. Jefferson, for instance, wrote, "The legitimate powers of government extend to such acts only as are injurious to others. But it does me no injury for my neighbour to say there are twenty gods, or no god. It neither picks my pocket nor breaks my leg."
(4) The doctrinally highly divergent Catholics and Quakers were present right from the start, exemplifying the actual practical commitment to religious liberty, not just the statement, and not just "either Protestants or something wishy-washy like Deism that we can pretend agrees with us". (Amusingly, the heavily-outnumbered northern Baptists anxiously wrote to President Jefferson for reassurance that he was committed to religious liberty. He responded in the affirmative.)
(5) That this was the case in 1787 does not mean that it has been for the entire history of the country, only that the principles laid out were laid out at least mostly honestly. For instance, the Second Great Awakening in the U.S. was a revival and intensification of Protestantism. If you read Maine's constitution, written at the height of the Second Great Awakening, the religious language is very prominent (https://www.maine.gov/legis/const/).
So, no, it simply was not a shorthand for "our brand of religious intolerance". It was an actual commitment.
Regardless, thank you for the link to the Center for Partnership Systems.