In the context of crowning kings, the union of the crowns was 1603, before the mayflower, before new Amsterdam, even before Jamestown.
But even with 1707 you’re talking 30% of the age of the USA again.
Between 1714 and 1807, Britain was also in personal union with Hanover. That didn’t make Britain and Hanover the same country any more than England and Scotland were the same country during the 17th century.
Between 1714 and 1807, Britain was also in personal union with Hanover. That didn’t make Britain and Hanover the same country any more than England and Scotland were the same country during the 17th century.