Are you talking about Puritans or Pilgrims?
frogjerk wrote: “Are you talking about Puritans or Pilgrims?”
On this point of wanting purity rather than tolerance for religious freedom, there's no difference between the Pilgrims and the Puritans.
The difference is that the Pilgrims were Separatists who believed the established Church of England was corrupt and secession was (depending on the Separatist writer) either permissible or mandatory, while the Puritans regarded themselves as being members of the Church of England who were still trying to work within it to purify it.
There is no way to argue that any of the early settlers of the colonies other than Rhode Island, Pennsylvania, or (in a very limited fashion) Maryland wanted religious freedom. William Penn wanted freedom for his Quaker views and was willing, based on his pacifist principles, to refrain from persecuting most other groups that today would be considered Protestant evangelicals. The founders of Maryland wanted toleration for English Roman Catholics and as a practical matter were forced to accept Protestant worship in their colony as a condition for obtaining toleration for their own views which were unacceptable in the home country. Only Rhode Island's founders affirmed religious toleration in a form resembling anything we'd understand today, and they did so in reaction to opposition and in some cases outright persecution by both Massachussetts and Plymouth colonial officials.
Neither sect believed in any sort of religious liberty. Roger Williams was driven to Rhode Island because of his beliefs. Mistress Anne Hutchinson had to find refuge in Dutch New Netherland.(remembered by the eponymous Hutchinson River and Hutchinson River Parkway) before moving on to what became New Hampshire. In fact one colony started for purely commercial reasons was the most tolerant and that was New Netherland,until Stuyvesant’s assault on the Quakers, which led to the New World’s first populist demand for religious toleration, the Flushing Remonstrance.