The pilgrims had a particular situation and were in a particular world, compared to today's globalized world:
The pilgrims were English. There were often wars between England, the Netherlands, France and Spain: so English pilgrims were not sure of being well received in colonies of those countries.
The pilgrims only had a different religion, and they were not prosecuted as political ennemies: this means that as soon as they were far enough for the metropolitan colony of England, at this moment, the English King had no true reason to prosecute them far from England. They were in the East of the New World, a vast colony with few people at the moment (contrary to Canada, India later on) and no crucial resources (contrary to gold/silver in South America for Spain for example). So fleeing there was enough for the pilgrims