I would argue that your point has merit, and that perhaps the law needs to be amended to enable employers to quickly and accurately cross-check SSNs. For example, such a system is already in place for gun purchases. I have purchased 2 firearms recently, and in both cases they checked my SSN, name, address, criminal record, record of any domestic violence issues (which is generally not criminal but is in family court), tax records (liability), etc. etc. They did this literally in a few minutes.
Would this be so difficult to implement for employers?? A prospective employer could check the SSN against a database, and confirm that the name matches, the gender matches, the approximate age matches, the location in the country matches, and that the SSN is not in use by 50 other people.
Then, employers might not have much of an excuse.
If you're here, I ought to be able to hire you. Requiring that I spend time and money to do a variety of checks and, if I fail on one, I can go to jail, is ridiculous.
If the government cracks down too hard on these guys, businesses will just refuse to hire Hispanics altogether, legal or illegal.
The political party responsible for THAT fiasco is going to be out power forever.