I am in limited agreement. I support doing so absolutely. It would definitely reduce the problem, but I am not convinced that the opportunities for, well, private work in form of a transaction directly between the illegal and the end user would stop. I am also not convinced that there isn't an incentive when we have courts ruling that the children of illegals who don't speak English are legally entitled to translators on the public dime, and assorted other factors which would encourage them overstaying the point most of us would consider them welcome. Further, even if it reached a point at which there is little to no incentive to enter the country illegally, that doesn't mean that there is any incentive for them to go back either.
Two different problems. Kut is trying to get his finger in the dike and stop any further leakage.
Deny the ones that are here work and transfer payments/public assistance, perhaps criminalize (at least to the extent of fines) giving 'aid and comfort'. Make it painful to stay.