The American yearning to connect with a European ancestry is pretty unique, especially given the earnest patriotism and love for their country that Americans have. Is it the romanticism of a more ancient history than that of the USA which appeals maybe?
Although the history of white colonisation of the US landmass isn't the oldest story in world geopolitics, it does go back to the 1600s and 1700s which is farther than most of us go in our search of where our personal story started. Even a mass emigration like the one which followed the potato blight of the mid 1800s is now 7 generations ago. How far do most of us go back in defining our lineage, 2 generations? 3, 4?
On my own ancestry for example, I only have to go back three generations (great grandparents) and literally everyone then and before then is Irish, but I certainly don't see myself as Irish, other than having an Irish surname. There's no ethereal voice calling me back to "the old country". I do really like it there and I've enjoyed every visit and felt comfortable, and the Irish are an amazing people, but I certainly can't claim to be Irish. My parents and grandparents were all born in the Northwest of England or the West of Scotland, and cultural cues from there are really what's shaped me. Not the fact that some people who I've never met used to live in Cork.
I work for an American company and consequently most of my colleagues and friends are American. A good half of my colleagues claim Irish lineage, I think that's clearly the one to have if you hanker for a romantic back story! One or two of them poke fun at my Englishness (I think they've been watching too many Hollywood films where the English guy is the baddie, or too many Joe Biden speeches perhaps), and they like to josh that we're "natural enemies" since they're "of good Irish stock". Ironic that in all but one case*, I'm closer to being Irish than them.
Bottom line is, we're all mongrels to some extent, it's probably a little dishonest to pick and choose which bit of the bloodline you like.
(*My boss is a New Yorker, but his parents went over there from Ireland in the 1940s and he still has living relatives in Ireland, so I get that one).