"it is true that having a handy supply of illegal immigrants keeps the wages for jobs low." That's a shorthand way of putting it tat we keep hearing in England too - except that it gets applied to legal migrants from places like Poland as well.

There is a very important difference between these two things. The whole point is that by making it as difficult as possible to become legal residents (as our government is presently doing), we ensure a permanent population of illegal immigrants. And when that happens, those illegal immigrants will keep the wages low because they will work for a lot less than people whose status in this country is not in question. That's the one of the reasons we have so many illegal immigrants.