The interpretation in the Prager video is questionable in a number of respects. It gives an overall impression of the United States government as far more closely intertwined with religion then is supportable.

The American founders were divided on the role of religion in the United States, so to speak about them as a coherent body on the subject is incorrect. The video uses a quote from George Washington's farewell address, about religion being necessary for the maintenance of political prosperity, and misleadingly implies that Washington advocated state sponsored religion. People like Washington could see religion as essential to the United States, but still believe in voluntary rather than state established religion.

The claim that "for Jefferson and the other founders, religion was central to the entire American project" is particularly an issue. Certainly Jefferson mentions "Nature's God" in the Declaration of Independence, but even in that text this deistic vision of God is equated with the "laws of nature." There were founders like Patrick Henry who saw Christianity as pivotal, but again, this was not an unanimous view. The Prager video also mentions "countless examples where the government acknowledges its debt to God." I've written a bit on various documents related to religion in the United States, in the founding era I'd be a bit skeptical of the existence of "countless examples."

That said, the idea that the First Amendment forbids establishment of a national religion is fairly tenable. The video is not incorrect that some states continued to have established churches after the Bill of Rights was ratified. It's also correct that the legal interpretation of the First Amendment as "separating church and state" was only definitely established in 1947 in the Supreme Court case Everson v. Board of Education, when the establishment clause of the First Amendment was applied to the states.

Overall, the Prager video is more an example of fringe interpretations that would be outside of the historical mainstream than it is an example of outright invention.

Recommended Readings on Church and State:

Davis, Derek H. The Oxford Handbook of Church and State in the United States. Oxford University Press, 2010.

Dreisbach, Daniel. Religion and Politics in the Early Republic: Jasper Adams and the Church-State Debate. Louisville, KY: University Press of Kentucky, 1996.

Fea, John. Was America Founded as a Christian Nation?: A Historical Introduction. Westminster John Knox Press, 2011.

Hamburger, Philip. Separation of Church and State. Harvard University Press, 2004.

Sehat, David. The Myth of American Religious Freedom. New York: Oxford University Press, 2011.

Note: Edited for clarity and spelling.