Don't deport every last one of them --- some would be too hard to find --- but deport many more than are being deported now. Even in this town where deportations commonly occur --- no Americans care, pictures of Mexicans walking back over the bridge after being let back into their own country through that gate by the INS, don't raise an eyebrow. If they do print a photo of it, it doesn't raise too much interest.
I have found, like you, that usually there aren't any group of people more against illegal immigration, than legal immigrants. They are hurt the most by it.