I have never, ever read in any book or heard anyone teach that the Mayflower was the first ship American to the colonies. So where exactly have you heard that myth before?
Statements can be held to be true through omission. I sincerely doubt that only a small percentage of Americans has any idea that the Mayflower was not the first ship. Look at this page from PBS.
http://www.pbs.org/wnet/historyofus/web03/segment1.htmlThe text reads:
Freedom in America has meant many things: freedom of speech; freedom from unfair, autocratic government; freedom to strive after personal dreams. But in the eyes of many, the fundamental American freedom has been the right to worship or not worship in any way a person chooses. It was supposed to have been that way from the beginning, Thomas Jefferson believed. He wrote: "Our forefathers left their native land to seek on these shores a residence for civil and religious freedom."
Back in seventeenth century England, religious freedom didn't exist. You had to belong to the Church of England or suffer persecution. King James I, speaking about religious dissenters, said, "I will make them conform themselves, or else I will harry them out of the land."
In 1620, 102 brave souls clambered aboard a little ship—named the Mayflower—and began their sail westward to the new world . Some of them called themselves "Pilgrims," because they were on a religious journey; they hoped to build a new society in America more perfect than any on earth."
It wouldn't be at all clear that others had already landed. The web page then goes on to say>
"Did you know:
n December 1620, the Mayflower was in Cape Cod harbor. Before the Pilgrims disembarked for good in Plymouth, Mistress Susanna White gave birth on board the ship to the first English baby born in New England, a boy she named Peregrine."
The key phrase here is "New England." Other English babies had already been born in the colonies.
The point here is that there's a mythos of New England primacy, which is functioning to erase the actual history of Virginia's primacy. This was done after the Civil War, victor's justice, victor's history, to pretend that the South had no part in the forming of the nation.