In America we are taught to view the Pilgrims as adventurers who endured great danger in order to seek religious freedom. The full story is, of course, more nuanced.
Many Pilgrims were Puritans. They did not sail to America to establish religious freedom for everyone, but rather to practice their own fundamentalist religion exclusively.
And they didn't go to America because they were thrown out of England. They first went to the Dutch republic, where they stayed for over a decade. They decided to leave Europe for the Americas, of their own free will, in no small part because tolerant Dutch society was proving too alluring to their children and leading them away from the cult. Thus they made the decision to move to a place where they could brainwash their children in peace without the inconvenience of regular reality checks from interactions with non-puritans.
There was actually a notable Puritan who supported freedom of religion. Roger Williams, founder of Rhode Island. He was a Puritan minister and Massachusetts Bay Colony threw him out for preaching about the idea.
He left for a bit but then went back to Massachusetts Bay and continued his preaching. The Puritans told him that he needed to leave under threat of death, so he went and founded Providence Plantations on the principle of religious freedom.
