It's true that the heritage of an American doesn't really matter after a space of time. I myself am German, Austian, and Italian, but I'm no different from my friend who's purely British and Irish. I think the point is that the culture as a whole has been influenced by more than just the Brits. Most Christmas traditions are German, and Americans celebrate a lot of other holidays. I know the people who take Spanish at my school always make a huge thing out of The Day of the Dead (or whatever it's called). Crèvecoeur was the one who first called America a "melting pot", and even though the politically correct term is apparently now "salad bowl", it's obvious that Crèvecoeur was the one who was right all along. America isn't just an extension of Britain; it's a mixture of all the cultures that have come to America, and we've formed an entirely new culture of our own. Yet, for some reason, we're all still proud of our heritage. It's pretty rare that you'll see a difference unless someone is very VERY proud of their nationality… like me.
Something I've always been a little skeptical of. I've travelled somewhat in the US and met a fair few Americans, but while most claim heritage of a good half dozen places unless the immigration was within the last generation or two then I've never seen any real cultural differences, although this is purely anecdotal.