The Pilgrims were Puritans who were dumb enough to sail on the Mayflower and land in Massachusetts right before winter.
They were separatists who had previously left England for The Netherlands. Not finding the paradise they wanted there, they decided to retain their English identity by sailing to North America to create New England.
Cue witch trials and banishment for those who weren't strict enough. Anne Hutchinson was a mouthy woman who dared to speak and teach in church and got banished to Rhode Island. All the bad kids got sent there. Sunday sermons featured family favorites like "Sinners in the Hands of an Angry God." ( God thinks you are disgusting and you will burn in hell forever for it. )
No, there was no freedom of religion. You either towed the line or you were banished, pretty much to the wilderness. It was those banishees who were interested in the separation of church and state. The church was the state in Puritan New England.
Ugly people, those Puritan Pilgrims.