The teacher, at an all boys Catholic military school, insisted that the challenges the Puritans faced in Britain were largely of their own making. She said there was little in the way of societal pressure or persecution for the Puritans in Britain. Instead what they faced was a country which simply didn't care for their way of life and resumed a secular life. This persecution complex eventually led to the Puritans seeking to found Plymouth and spread from there where they could finally be in power again and be free to fully persecute anyone who refuses to adhere to their lifestyle. My question I suppose is two-part, how accurate was my teacher's portrayal of the broad generalizations that led post-Restoration Puritans to flee to the New World? The second part has this perceived persecution, or persecution complex, traveled from those early Puritan settlers through all the great eras of American history?
/r/AskHistorians has a great answer
Your teacher has it pretty close, the Puritans were basically religious extremists, but there were also non Puritan colonists mixed in with no particular religious zeal. It got pretty contentious between the "Puritans" and the "Strangers" as they came to be called.
I want to address some of the other things in your post outside your specific questions at the end, because you've detailed a pretty bizarre assortment of claims made by your history teacher.
For one thing, they don't make chronological sense. From your post, it seems as though your teacher is arguing that puritans, after controlling the English state between 1649 and 1660 and aggressively persecuting non-puritans, left in droves for the colonies after the 1660 restoration of the monarchy in order to refound their persecuting state. Here's the problem. Plymouth was founded in 1620, Massachusetts Bay in 1630, and the major wave of migration to New England was in the 1630s. The Restoration of Charles II to the thrones of England, Scotland, and Ireland occurred in 1660. So the Restoration can't be the causal impetus for the foundation of the New England colonies, because they were already extant by the Civil Wars, let alone the Restoration. Indeed, only two years after the Restoration, the Halfway Covenant showed that the New England churches were, if anything, moving in a more inclusive direction by that time.
For another thing, the idea that puritans did in fact control the English state from 1649 to 1660 and were aggressively persecuting non-puritans doesn't reflect the facts terribly accurately either. In one sense, the 1650s were actually more tolerant than the decades on either side, since Protestants of all stripes were generally given a pretty wide latitude to practice their faith. The state did, in this time, launch some intolerant measures (proclamations against games and Christmas, suppression of Baptists and Quakers, and even some executions of the latter in Massachusetts), but the pre-Civil Wars state probably would have suppressed the explosion of religious sects in the 1650s more vigorously than the Commonwealth and Protectorate did. Moreover, the people who were in charge of the government reflected a pretty broad range of Protestant religious belief, and defy easy grouping as "Puritan."
There's also the problem that practically nobody would have self-identified as a "Puritan"--"godly" was a favorite term instead. "Puritan" was a term of insult, which suggests at least some level of social pressure against these "hotter sorts" of Protestants, but also unhelpfully tends to describe a vast quantity of people. The Brownists who founded Plymouth were subject to even more extreme pressure, which was why they ran away to the Netherlands first. I think it is not unreasonable to suggest that by the 1620s or so, persecution was more a story about their own identity puritans told themselves than something firmly grounded in reality. But even then, there was only a respectable puritan core who were fundamentally accepted by the existing establishment. Any deviance to the degree of, say, the Brownists, who felt it necessary to flee the Netherlands before founding Plymouth, or Richard Wightman (last man burned at the stake in England)., wasn't tolerated. So speaking about them collectively is always tricky.
Lastly, I have never seen any serious scholarly suggestion, anywhere, that the people who founded the New England colonies did so with the aim of being able to persecute others who did not share their faith. Certainly the New England "saints" ended up persecuting people with perceived heterodox religious views after the colonies were founded--that's why we have Rhode Island, for example. But they did not sail across the Atlantic itching to found societies for the purpose of persecuting those who disagreed with them. They sailed across the Atlantic because they tended to believe that conditions in England did not permit the founding of their ideal godly societies. The "New World," on the other hand, offered a blank slate (as long as one did not count the Indigenous peoples there, which these English immigrants did not).
Your teacher, therefore, is right to suggest that the puritans who founded the New England colonies largely did not do so to escape active persecution. But the process your teacher's used to come to that conclusion is terribly flawed, and has produced some shockingly bad history along the way.
Sources:
Ian Atherton and David Como, "The Burning of Edward Wightman," (2005).
John Donoghue, Fire under the Ashes: An Atlantic History of the English Revolution (2013).
Robert Harkins, "Elizabethan Puritanism and the Politics of Memory in Post-Marian England," (2014).
Carla Pestana, "The Quaker Executions as Myth and History," (1993).
Carla Pestana, The English Atlantic in an Age of Revolution, (2007).
Michael Winship, "Were There Any Puritans in New England?" (2001).
Michael Winship, Godly Republicanism: Puritans, Pilgrims, and a City on a Hill (2012).
Blair Worden, The English Civil Wars, 1640-1660 (2009).