They want the illegal immigrants because they are nominally Catholic. If they were Protestant I doubt the Bishops would be for it.
Sadly, if they were muslims they would bring them in by the bushel load. Catholic Charities has done that around here (along with the ELCA’s Lutheran Charities).
I disagree with that assessment. As you say, many of the immigrants are "nominally Catholic." I expect fully a third to convert to Pentecostalism or other branches of the Evangelical churches shortly after arrival. Hispanics are a socially conservative bunch and makes for a natural conservative constituency until alienated - no pun intended - by people afraid of their "cooties" (I'm not talking about you).
The issue is far more complex and its description cannot be reduced to soundbites easily.
~Theo