Exactly so. In fact, I recall the Dept of Labor was going after employers for refusing to hire illegals.
If it is the same case I am thinking of, the Dept of Labor went after the employer for demanding more than the statutory amount of identification. It is not lawful to hire illegal aliens without work permits. The Dept of Labor knew exactly what they were doing: promoting illegal immigration! But they would claim they were simply arguing the law.
One thing Trump could do would be to reform the I-9 system to give employees far more latitude in determining whether an employee is legal or not. This is little rope-a-dope of pretending to enforce the rules, and big companies pretending to be diligent about following the law, needs to go. Employers should be able to use fingerprints, ID, utility bills, references, background checks...whatever, to establish identity.