hello; A simple answer, tho perhaps not a best answer, is to let the illegal parents decide about the children. The parents made the decision to sneak in, let them decide what is best for their kids.
Hello; Yes, and again the parents made a decision to break the law. The parents themselves created the mess. The laws have been in place a long time. As far as I am concerned the parents own the mess. If they can work out a way for the citizen kids to stay in the country, so be it. If not, take their children with them.
Hello; I follow the logic on us having to pay for the care of minor children who happen to be left behind. To me that should be a last resort in the process. Let the parents work out some sort of stable life for the children they leave behind if that can. If the parents cannot or will not work out a place for their children to stay then they make a choice of either take them with them or abandon the children. By the way the children will be citizens and can come back when they reach majority age.
The abandoned children likely will go into a system and may be adopted because as far as I am concerned the parents should lose their rights as parents.