First, I'd be shocked if President Washington was a progenitor of mine (since he was childless). But yes, I would be proud of ancestors who accomplished such noble and wonderful things. Same for Ben Franklin or Edison. But I wouldn't consider myself superior for that fact. And if my ancestor was some unaccomplished president I would find that an interesting fact but nothing more.
By the way, my great-great-grandfather was George Washington. But he was not related to the first president.
These were great men, of course someone would be proud if one of them were their ancestor.
However, the British royal family and what is left of the aristocracy are not necessarily descended from people who were great or brave. Instead they are descended from people who like them were born into the club, not people who actually earned their status and fortune.