This sounds like a just so story. There were all sorts of groups who set up shop in America, and all contributed to its success and influenced the culture.

Virginia was the most populous colony during the revolution, did English planter society just disappear and the Puritans made it all the way down to the South?

What about the Quakers in Pennsylvania?

Dutch society in New York?

Poor Scots in Appalachia?

And, in any case, this campaign started in Australia. Were there a lot of Puritans there?