Regardless of if they achieved their religious objectives, that earnest mission to make every human soul capable of reading the Bible for themselves produced the social good of a literate population capable of reading prodigious amounts of non-Bible content.
It is an interesting consequence of how the religious wars in Europe spilled over into in the early Americas.
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The Puritans were always few in number and were demographically displaced by later immigration around the fishing industry in New England.
I think some overstate the influence of Radical Protestants on American ideology with offhand references to Max Weber or by calling whatever their pet cause is a fight against "secular puritanism." On the other hand, I do think there are some interesting parallels.
For example, one could argue that the mistreatment of colonists by the mother country was overstated by a population already distrustful of the Crown. I'm no expert, but it would be interesting to read more about that dynamic.
But it surely did happen -- IIRC, Adams and Jefferson were both noting in their correspondence how by the end of the 18th century most of the Puritan descendants had somehow become Unitarians.
They were influential in a narrow geography of the Massachusetts Bay Colony for about 50 years. Their own children and grandchildren largely rejected Puritanism resulting in the Half-Way Covenant and the eventual demise of Puritanism. I agree that they're part of the foundational myth, but it's just that myth.