I am not saying the child of a union between a citizen and an illegal alien should go anywhere. My suggestion is that the child of that union should not get automatic U.S. citizenship.
Since the child is the result of a union between parents who are citizens of different nations, you could make the argument that the child is a citizen of the nation of the illegal alien parent. I think the illegal parent should be deported immediately and the child should go with whichever parent is better able to provide for it.
I don't agree with your suggestion to end all immigration, only illegal immigration. Immigration is necessary and desirable. I just think it should be controlled. By having a process for immigration, we can vet who is coming into the country, have a set of requirements, and we have a better chance of assimilation into our society.
However, anyone caught illegally entering the country should immediately be deported the same day. No more catch and release within the U.S.