I think a lot of it's that conservatives and Evangelicals and other sects of Christians claim the monopoly over the American culture and identity. Liberal and or areligious Americans, especially white ones, don't feel like they have a culture. It's not true of course, but it's not as glamorous.

Like you said though, Irish American is a totally separate culture than being Irish. Even then, people who are Irish American aren't necessarily in the quintessential Irish American culture. That's where you get suburban Massachusetts girls named Shannon taking a gap year in Ireland to find herself and working class dudes in Southie who are as Irish American as they come but would tell you to go fuck yourself if you asked him anything about the island.