What would be conservative about that?
The Puritans and Pilgrims didn’t have “religious marriage” at all, it is apparently a newer idea than that. Marriage between a man and a woman going back thousands of years has the been the bedrock of civilization.
Giving that up would not be “conservative”
Puritans were dissenters from the Church of England which, like the Catholic Church, treated marriage as a sacrament. So, Puritans were in the context of their times the “radicals” in this matter and the Catholic Church and the Church of England were the “conservatives.” Now, you can go further back in time and argue how many “true” Christian sacraments there are. Some say only two: baptism and communion, but that is another argument.