I've never been quite clear on this. We lionize the Pilgrims, but they were one specific bunch in one colony. As far as I can tell they didn't last long as a sect. Many people claim descent, but none are actually Puritans. (Though they may well be lower case puritans.)

Many of the vaunted Founding Fathers were deists, which is just this side of agnostic. They were probably also puritans but none were Puritans.

As far as I can tell we reinvented religion during various Revivals, and what we came up with has only the faintest connection to the version of Christianity that people came here with. They claim connection to that one particular sect, but without either genealogical, intellectual, or theological descent.

I don't fully understand the actual history. But much of what we claim about it is a myth.