Probably not most Americans. I think once people are 3 or 4 generations removed from their most recent immigrant ancestor, their ancestral origins become mostly a curiosity. Some people get really into documenting ancestry but it's usually just a hobby.
All 4 of my paternal great grandparents were immigrants from Italy, and I'm even a dual US/Italian citizen, but I don't feel any real cultural connection to Italy . I would never call myself Italian or identify as Italian-American.
My mom's side is various flavors of British and her family has been here for hundreds of years, but there's pretty much no feeling of fondness for or a bond to the UK as a result of family history.