The mathematics of genealogy fascinate me. Like you say, almost anyone with any English blood can find their way back to one or another of the kings, and moreover, if you had an ancestor in this country before 1850, it is a virtual mathematical certainty that you are a Mayflower descendant as well.
What's more, it can be shown that essentially all people of European ancestry are descendants of both Charlemagne and Mohammed, and practically everyone save a few pure-blood Indians and Pacific Islanders are descendants of Julius Caesar, Nefertiti, and Alexander the Great.
There was a great story on this in the Atlantic Magazine a few years ago, called "The Royal We".
-ccm