There are notable issues in the Southwest in the United States where people don't speak any English at all, and speak only Spanish, suggesting they are legal citizens born of illegal immigrants, or are still illegal immigrants themselves.
"When I hear people say this, it's not because of any experience they've had, but just a repetition of talking points"
You pointed out a potential problem. So my followup is: "What negative experience have you had because of this?"
I'm not claiming integration problems don't exist (see my original comment). I'm claiming they are very minor, and blown way, way out of proportion. So much so that while I've known many who complain about the integration problem, I've yet to find a single person who's had direct negative experience related to this.
And, as an aside, give people in their 20s a bit more credit!