When ICE raided Tyson Chicken (a few years ago), multiple workers provided documentation from Tyson telling them how to stay under the radar and how to fill out paperwork if they were undocumented. There's definitely a very large effort in undocumented labor... and little interest in rocking the boat of those employers.
Yes. The USA runs on undocumented work in many ways. This is a far, far, far (etc) cry from government-funded citizens escorting around and signing up for lines of credit and otherwise paying the way for undocumented workers.