I don't know about that. They stay here because someone is paying them - from the meat packer, to the hotel to the restaurant - they easily find gainful employment. You cut off all social services and really slap horrifically punitive penalties on ANYONE who employs an illegal, they'd leave.
When you talk "deportation", the media does a pretty good job of painting the visual of millions of illegals packed into cattle cars, hearkening back to the Holocaust. That turns-off independents.
Republicans should turn their back on the corporate masters, and pass some laws that really screw these employers.
There are a lot of FReepers whose ideas run precisely along those lines. The practicality of rounding up illegals, and the political and legal realities of dealing with the people so-collected, are lost on the border-bots.
You are quite correct that the problem originates among those who are paying illegals to come here. I would extend the blame to various state, local, and federal entities that provide long-term welfare benefits to those folks.
Oh, in practicality that is a better way to get them to leave than to try to deport every one of them. But Newt’s saying we’re not going to enforce our laws by either de facto or de jure deporting them, that we’re going to let them stay. And if you let them stay you are offering them amnesty.