There are a billion people who are descendants of Charlemagne (essentially, one third of the world’s population) and one out of every 8 Americans is a descendant of one of the Mayflower passengers. Descent isn’t that big a deal, and before DNA, proving it was! And, BTW, there were several Robert The Bruces.
If you have any ancestors who lived in Massachusetts in the 1600's or 1700's you are even more likely to be a Mayflower descendant. Very few people lived in the colonies and the families intermarried. It doesn't mean you're descended from royalty though, 20% of the passengers on the Mayflower were indentured servants.