Okay- how will we identify people as illegal? Universal identity cards for all citizens and legal immigrants that will need to be carried at all times is the only way I can imagine. And once we work this question out, then do we raid certain businesses, all businesses, certain housing, all housing, or just stop people on the street at random? And of course, who will judge if someone is legal; a court or a police officer? Where do we put the illegals once caught before their plane or bus leaves (it will be a lot of people)?
In fact, shouldn't we really start the whole process with your idea of a Constitutional amendment to declare that children of illegals born in the USA are not citizens? Otherwise, we will have a bunch of underage citizens in the USA after their parents are deported to other countries.
No, you are right- it is pretty simple. Simple to say; a lot harder to do.