There was still plenty of room left in Europe when pilgrims settled in America.
You're assuming that the task of crossing the Atlantic in the 17th Century is a feat comparable to a more advanced civilisation travelling dozens of lightyears in space. We are a more advanced civilisation - and not only are we still doing pretty badly at human space exploration, we're staring to form pretty successful scientific theories that show the task will be very, very difficult - and could be impossible. You're basing your argument on the (non-falsifiable) notion that an advanced civilisation will develop technology indistinguishable from magic - in an age where science is capable of asking quite a few awkward questions about magic w.r.t. little things like causality and the laws of thermodynamics...
At that time, travelling to America may have not been a picnic, but was still "only" a matter of months. Ships were readily available (the Mayflower was just a garden variety merchant ship). Coming back was unlikely (for the majority of the passengers) but not impossible. Trade with the old world was still feasible (much of the exploration of the new world at the time had a view to bringing resources back to Europe) and the climate on the East coast of America may have proven to be a bit nippy, but you could breathe the air, drink the water, eat native plants and animals and be reasonably confident that your seeds would go.
So, the question is, would the pilgrims still have left Europe for America if it meant a shipbuilding programme that made Apollo look like a science fair project, then spending the rest of their life on a ship, never seeing land, in the hope that their great-grandchildren would finally arrive in America - and then face the task of another generation or two on the ship terraforming the land before they could start ploughing and planting?
Especially given that, if you could buy a ship that could survive for many lifetimes in the middle of the Atlantic without support, wouldn't it be a hell of a lot easier just to build a big raft and park it sufficiently far offshore that the people you were running from wouldn't bother you?
Then, seriously, what do you think the chances are of a bunch of religious fundamentalists crewing a generation ship without overpopulating, schisming, squandering resources, killing each other and regressing to savagery (the 56th law of Science Fiction)? Yet in a society without the tendency for people to persecute each other in an argument over the colour of the sky fairy's wings, their motivation for embarking on the journey wouldn't have existed...