This is another issue. I don't think they should be automatically made U.S. citizens unless one of their parents is laready an American.
The rest of your analysis is pretty heady stuff. One false move on Bush's part brings the whole game to an end. And, we lose. Plus, what about the law of unintended consequnces? Happens all the time whe bureaucrats get involved. Who knows what these knee-jerk reactions to problems pose to our progeny.
You would need a constitutional amendment to change the fact that children born in U.S. have U.S. citizenship or dual citizenship. And no one viable will support changing that.
What may be done is to change the law with regard to dual citizenship, so that a child born here who is taken back to the parent(s) native land, would have to decide at some point after coming of age which country they belong to. And there should be some law on how that decision is made or how it is made, de facto, by the actions of someone living in another country who was born here.
Then there are the illegals that have children here and get to stay BECAUSE they have a child here. This is for down the road, but we can work to get to a point where we could make the law state that the parent has to go back (be deported) and take their American-born child with them. The child then has dual citizenship, which would be decided later as above. Or they can abandon the child and the state can put it up for adoption.
These will be toughies to get passed, so we will have to build our majorities before we can hope for such changes in the current law and practices.