No, we just stopped being Polish, Italian, German, Irish, etc. and became Americans like the rest of the population...
No, I don’t think it was solely an immigrant thing. Catholic kids went to Catholic schools not only in New York and the big ethnic centers, but all over the country, including places where their families hadn’t been “immigrants” for over a hundred years.
There might have been some decline in urban areas, but certainly, the “old” immigrants would have been replaced by new ones, such as Hispanics, if it was simply a matter of ethnic identity.
What happened was that the schools changed. Most of them are no longer staffed by nuns or religious orders, which changed the environment and also made the schools more expensive. Most of them abandoned Catholic doctrine and were distinguishable from public schools only by the slightly better behavior of their children, whose parents obviously cared enough to send them to Catholic school. I had my kids in Catholic school years ago and took them out because I figured it was better for them to hear bizarre leftist pap from a public school than from some place where they might make the mistake of thinking it was Catholic doctrine.
At my local parish school, the majority of the kids (and lots of the teachers) are not Catholic. Catholics feel no urgency to send their kids to Catholic school because in most cases there is no longer anything specially Catholic about it.
Furthermore, the inherent racism of the liberal mindset is on display. Catholic schools in black areas, for example, often attracted non-Catholic students whose parents wanted to spare them the dangers of public school (and large scholarship funds were set up to pay for them). But no effort was made to convert these children and their parents, and in some schools that I knew, the kids did not even have to participate as spectators in the religious activities. That was during a disastrous period when the Church stopped preaching the Truth; now, at least, in most schools parents do have to agree that their children will participate in religious activities.
The problem was not “becoming American.” It was the liberal takeover of the Catholic Church, which destroyed the religious orders and took away the heart of the Catholic school system. There was no longer any point in sending your kids to Catholic school, and that was what killed it.