I'm a contractor, and I can tell you that that commonly held agreement is way overblown. Sure, illegals will work like the dickens, but most of what they produce is sub-standard crap.
I've been dealing with the results of cheap illegal labor for twenty years in my trade. I don't know how many jobs we've made good money on, where we were hired to correct the schlock work of illegal construction workers.
My cousin had the same experience. Ridiculously shoddy work.