During the last depression there were illegals in this country - they left for the most part when the economy went belly up. Fast forward, its not happening. You would have to have congress address those issues that keep them here and keep them coming. Welfare comes to mind, medical, dental, public school system, judicial "assistance", free college tuition, illegal drugs, too. Also, the border was patrolled and if someone fired on border patrol back then, no one would have been sued if they fired back. Horses were employed for a great deal of border patrol, too. Jobs for US horses...gone.
Before massive federal taxing and resulting wasteful spending and unbridled power of congress, individuals could hire individuals to do work for them. Maids, gardeners, nannies, etc.,. Not "rich" individuals, just the plain old middle class. Today, the average individual can't afford such help, even though both spouses are working, b/c congress has decided they know better than the individual how to spend the individual's money - and to whom to give it to. And, they have made it "seem" as if people that do such jobs (maid, gardener) deserve not to, unless ... and only unless they do so for a very wealthy family. So, blaming it on large corporations is not entirely fair. I don't remember corporations lobbying for federal taxes - the fed, yes..of course, but not Chryster or Ford, Bell, etc.,.