If there truly is a labor shortage, not just a demand for low cost, benefits free labor, then increase the allowed number of temporary workers in the various immigration worker classes that already exist according to need.
Go after employers who are cheating the tax and social security systems. It's sorta the same logic they use in the war on drugs. Go after the source of the demand, the user, or in the case of illegal immigration, the employers.
Those employers who cheat by hiring illegals make it harder for those who play by the rules. They increase the number of illegals taking advantage of "free" health care, education and other benefits that should be reserved for CITIZENS or immigrants who are playing by the rules and that raises the price (taxes) for the rest of us. It's circular, b/c the increased price (taxes) makes hiring illegals even more attractive. A vicious cycle.
“If there truly is a labor shortage, not just a demand for low cost, benefits free labor”
Where I live, people with a high school education stand in line for a chance to get a physically difficult $9.00/HR job with medical benefits.
There is NO shortage of people to do these “jobs that Americans won’t do”.