"The huge cost of illegal immigration is the cost of the social safety net the US provides to the progeny of illegal immigrants over successive generations , chiefly education and health care."
I agree that switching to a national sales tax will not eliminate the costs of illegal immigration, and that we should amend the law (or, if liberal judges strike it down, the Constitution) so that automatic U.S. citizen is only granted to children of U.S. citizens or U.S. permanent residents (not of illegal aliens or tourists). But a national sales tax will create a disinsentive to illegal immigration and collect more money from them than the income tax does. There aren't any silver bullets when trying to solve the problems caused by illegal immigration, but a national sales tax could be part of the solution.
But a national sales tax will create a disinsentive to illegal immigration and collect more money from them than the income tax does. How? You pay an illegal $1000 for some service, are they going to remit a tax? When they purchase goods legally today, they already pay embedded taxes in the costs of the goods to cover all the taxes of every employee and business owner in the chain. Besides a good portion of that money is going to be sent back to Mexico anyways, so neither system could capture it.