Two factual correctives.
First, it is readily apparent to me, as a fairly closer observer of immigration debates state and national, that just about everyone uses the phrase “nation of immigrants.” Most restrictionist speeches I hear start out with something to that effect. It is not a marker of an open-borders position. It may have become a trite and useless phrase, but that’s a different problem.
Second, if you think that Middle America doesn’t continue to revel in nation-of-immigrant lore, you need to get out more. Go to Oktoberfest in various Midwestern towns. Visit Cajun Country. Watch who rides in small-town parades. Go to the Highland Games in North Carolina. Actually, go to certain nooks and crannies of the North Carolina mountains and hear some very interesting, often strange dialects and sayings mutated from the original Gaelic.