The Pilgrims came to America for religious freedom and to avoid persecution.
Half-truth, at best - they wanted the freedom to apply their religious more stringently - sort of an English Taliban. They were hounded out of England and weirded out of the Netherlands because of their puritanical practices - hence the derogatory Puritan label.
They wanted to establish a community in what is now the US to be LESS tolerant than the communities they had in Europe.
So - did they come for religious "freedom?" Yeah, kinda - as long as it was their religion, specifically. They definitely did not come in the spirit of "all religions should be treated equally," which is how this usually gets glossed over in US schools.