That is a mystery of the ages which I, an admirer of Traditional New England and the Puritan/Federalist heritage, have often wondered about.
By 1701 Harvard had already become unorthodox (which is why Yale was founded). John Adams was already a Unitarian, and Benjamin Franklin has been accused of everything from deism to satanism.
It is also true that the radicals of the early federal period (radical abolitionists/women's rights/free love advocates) were of old Puritan stock (though there were Jacksonian radicals as well). Such figures as William Cullen Bryant and William Lloyd Garrison actually started out as conservative Federalists, and anti-rationalist Fundamentalist Presbyterian Lyman Beecher begat a whole family of unorthodox radicals.
What is even more confusing is that the Republican Party was originally the organ of these people (to the extent that neo-Confederates accuse it of being "red from the beginning" and a tool of German Marxists and home-born radicals, and Horace Greeley is often considered an Insider by JBS types). Then it gets even weirder. Some of these really radical types then turned against the Radical Republicans for being corrupt and began calling for leniency toward the South (Charles Sumner was actually one of these people, believe it or not). They wound up allying with ueber-conservative Democrat Grover Cleveland, and later in life some actually became more radical than ever. BTW, would you believe that America's oldest radical publication, The Nation, was originally founded by conservative Grover Cleveland Democrats?).
The history of American ideology is truly a riddle wrapped in a mystery inside and enigma. My own personal position is that much as I admire the Puritan/Federalist tradition (and defend it from neo-Confederates and Jeffersonians), I don't get my ultimate beliefs from them.
Ultimately, the United States was a product of "enlightenment" thought. It was inevitable that some of this stuff would begin to happen eventually.
I forgot to mention that if you go to ultra-fundamentalist Primitive Baptist web pages (yes, they do exist), you will still see them invoking the Puritans and even Oliver Cromwell on some issues (though they disagree on others, such as church and state).
Don’t know if you have already read it (you sound like you are already well read on the subject) but I’m reading a book called “The Cousins Wars” by Kevin Phillips. The basic premise is that The English Civil War, American Revolution and US Civil War were all different stagespart of a contiguous Anglospheric Civil War in which the combatants shared a similar religious, political outlooks and even, to a certain extent, shared ehnicity (Parliamentarians, Patriots and Yankees being disproportionately of East Anglian stock and Puritan/Conregationalist), and the Cavaliers/Loyalists/Confederates being disproportionately High Anglican or even Catholic, and descended from Scottish, Irish or settlers from the North and West of England. In other words, those parts of the British Isles most prone to Tory and Jacobite sympathies. Interesting stuff, and of course, the “same side” ended up losing all of those wars (including the Jacobite Uprisings).