If they want to leave Europe, the thing that the pilgrims required to do is: D: Pledge loyalty to the king in return for passage to the New World
But knowing that the king have very little influence in the new world, many of the pilgrims that manage to get out actually form their own government and disregard the initial deal. One of the most famous colony that made by these people was called the Plymouth Colony,
I am thinking its either because they were still learing about medicines or maybe they accidently killed a man while doing surgery or something.