Hey hey! I grew up with religous scholars for parents, spent my childhood in church, and study colonial American history - so I think I may be able to tackle this one.

The thing that bothers me about that explanation is the term "freedom" which implies for all. While they were far more tolerant, ironically, than their Puritan neighbors they really went to gain religious independence. They didn't care what religion the natives practiced - they wanted the ability to practice their own without hindrance or influence from others or their religions.

Puritans were a faction created to "purify" the church of its misguided ways and bring it back in line with scripture by their interpretation. A sub faction of Puritans was the Pilgrams. In other words all Pilgrams were Puritans, but not all Puritans were Pilgrams. The Puritans felt religion was truly democratic and without hierarchy, which the Church of England had. The Common Book of Prayer, for instance, told an individual how to connect with God which the Puritans felt was the duty of the individual to do. They did not support holidays like Christmas or anything not represented by/in the scripture. They were against hymnals being sung as well (which is still found in numerous denominations today while others have electric guitars at services).

Why then? In 1603 the Queen had died and King James took over. The "reformers" (Puritans and those wanting reform of the church) were hopeful James would allow individual congregations more latitude in scripture interpretation (essentially allowing multiple factions or denominations under the banner Church of England) but he instead enforced the rule of the Church even more harshly, promising to remove those who dissented against it. The Puritans that would become our Pilgrams had enough and in 1606 met in secret to form their separatist movement. At that point they were targets for oppression and William Bradford noted the watchful eyes that seemed to follow them in day to day life and the overwhelming fear of the impending, so they left.

So as a result of the divide the Pilgrams had become a separatist movement which is just what it sounds like - they wanted to seperate from that church, which was not allowed as church and state were intertwined (the Puritans were, of course, similar in their religious interpretation but were not separatists and that's the biggest fundamental difference between the two groups). The ones we speak of left England under fear and threat of persecution to land in Holland, which wasn't ideal and never really worked as a new home but allowed them more liberty in religion than back in England. Politics there, the exposure of their children to other religions and impurities, and for a few other reasons (like the Dutch-English truce ending and fear of English capture) they grew concerned about staying in Holland, so they sought an opportunity within British North America, allowing them to remain british subjects yet practice religion freely as they saw the bible instructing them to. They sought permission but were not granted religious freedom. They decided the Kings reach on religion certainly would not reach so far away and planned to send a portion of the congregation with the rest planning to follow later.

As far as where they were going, well it wasn't really meant to be so unknown. They had not recieved a colonial charter and were largely funded by a somewhat private joint-venture, The Merchant Adventurers, and given permission to settle within Virginia by the Virginia Company. They missed by a few hundred miles and wrote the Mayflower Compact as a result (since there was no law or charter where they actually landed). Their sponsors made some demands, among them that non-pilgrams would go as well. Some of those added had even crossed the Atlantic before, one being on Newport's ill-fated trip to Jamestown nearly a decade earlier that crashed in the Caribbean. He made his way back to England and sought another opportunity to try his luck in Virginia. So it wasn't so much the unknown as much as it was them being the wrong kind of people to form a fresh settlement. They weren't farmers or fisherman and their early efforts at both were terrible. Natives showed them proper planting techniques that historians generally accept were learned by those natives from the English in the first place. They just weren't equipped for the task at hand and 7 months of terrible luck exacerbated that a great deal.

As to your second question, that's highly unlikely. The pilgrams were a very small faction that had little sway in the protestant movements that hit early America. The Puritans held much more sway in New England for a lot of its colonial history, which may have had an even greater impact given the lack of tolerance for other religions they expressed, but even so it becomes a smaller portion of the whole but may have contributed some to the Great Awakening in America. I must say, though, I dont believe the premise of this question is correct.

The diversity of religion found in America is astonishing even when viewed solely with a Christian lens. We find numerous different denominations of the same faith in nearly all bodies of citizens, some more tolerant of liberal social policies and some more conservative. I wouldn't blanket label the conservatives "fundamentalists" though some do exist within that group. What led to this spread of religious beliefs was largely the evangelicalism and revivalism in the U.S., namely the south, which gave way to denominations like Primitive Baptists which were (are?) much closer to fundamentalists than current "evangelicals" are. Without getting too much into current politics, the ideal of being a fundamentalist and actual believing in fundamentalist beliefs are very different things particularly in politicians. Politics also tend to be dominated by previous generations and dating back to Socrates we find divides in tradition between youths and adults. There would be virtually no church willing to have a guitar in Sunday service 30 years ago, yet now it is common. The Church of Christ I attended as a boy had hymnals but no organ/piano. My grandmother would have been furious had she known we switched to a Baptist church that not only had both but also bell choir, violins, and a much looser interpretation of the scripture as it pertains to worship. My parents would never accept a rock guitar, however, which is rapidly growing in younger denomination churches. For all these reasons it appears as though the older folks are traditional or fundamentalists, yet that's just perspective from society which tends to move progressively over time. I don't offer this anecdotally but rather as a micro analogy of the larger picture in which society continually moves toward what used to be uncommon or more controversial which is evidenced repeatedly and continuously in our history. Politicians and laws just tend to be the slowest parts to change, perpetually leaving them in a more conservative (as in fundamental) position by default.

Regarding the second part of this question, asking about any possible links between New England Puritans and modern religion, it's worth observing that the contemporary denominations that trace their roots back to Puritanism are as far from Christian fundamentalism as you can find among American religious groups. The direct descendants of Puritan churches are either part of the United Church of Christ or the Unitarian Universalist Association. These groups ordain women, support same-sex marriage, see the Bible as a historical product rather than an inerrant text, and are generally understood to be part of the religious left. Most Unitarian Universalists no longer identify as Christians, and a sizable portion of the membership rejects theism.

All this is to say, there is no direct organizational link between the settlers of Plymouth and contemporary conservative Christianity. Some United Church of Christ and Unitarian Universalist congregations still met in congregations originally organized by the Puritans, but their theology is very different from their forebears. Religious groups can change dramatically in a few centuries.