The whole appeal to hiring an illegal is that he is "off the books" and willing to work at below market wage and without benefits. They will also not report any sort of workplace violations as they are illegal and afraid of being deported.
Put the illegals "on the books" and the appeal of hiring goes away for the overwhelmingly vast majority of illegals.
Sorry, but I know illegals who make MORE money than their US counterparts, because they are harder workers.
And the fact is MANY Americans are hired "off the books" for the same reasons..because it is mutually beneficial to both parties...the workers don't have taxes taken out of their pay.