The contractor or one person taking the check and distributing makes sense. BUT then the money trail goes to him, if his illegal people he pays don't have SS numbers, then he'd have to charge a lot more than 15% because if his money doesn't go elsewhere he'd pay tax on it.
The SSN seems it should be an easy thing for the government to monitor and fix. Wages should not and could not be coming in for anybody under 16 years of age since they can't work. If they see 8 year olds having thousands of dollars that should red flag as child labor law or illegal/stolen. If SSN is of working age and 1099 comes in, and the illegal doesn't file taxes and pay SS tax on it, then they would find the person who is issuing the 1099 to track down that person. Unless the SSN only have less than a taxable income.
About home prices going up if illegals were gone, I disagree on that. Home prices are way inflated. Home being built with cheapest materials and cheapest options, costing 40 or 50k to build, then selling for 250k etc. Or the contractors charging over 100 per hour and paying just $10 an hour to the labor, etc.