The difference was the wretched refuse went through Ellis Island, went to work, and embraced being Americans. They weren't looking for a handout, or to change their new home, they wanted to change themselves to their new homeland.
And as far as having kids here. I'm fine with them being American citizens, but that doesn't mean their parents have to stay here. When I was in Germany, children born to American servicemembers were German citizens. That didn't mean that they and their parents could automagically stay in Germany. (But they were still subject to Germany's compulsory service when they turned 18, unless they renounced their citizenship.)
They also weren't automatically US citizens, their parents had to fill out paperwork and send it to the US Embassy in Bonn. Which would take anywhere from weeks to months.
So I have no problems deporting the parents, the children can either join their parents and come back when they are 18, or they can be left behind with legal family members, relatives, or others.