A great idea overall! But I do quibble with your second point. The parents are already criminals for invading the boundaries of the Republic. Therefore -- without exception -- each and every set of parents is, by all but a leftist view of the law, unfit.
I don't care one iota about "breaking up the families" since the parents are criminals from the get go. The children would do far better with new parents, preferably Christian (or Jewish) Conservatives. They'll raise 'em right! And they'll speak English!
Why do you think we should believe illegals would not think leaving a child behind in America is a good thing.
We are told in literature and film that poorer mothers often give up their babies to others for a potentially better life. If we consider the natural impetus of procreation is to perpetuate genetic traits to the next generation, then the Illegal Alien has won the lottery of life by dumping their progeny here. The child is not ours to care for, so they must leave our country with their criminal parents.