You have hit on the major problem with DAPA and DAPA; the one no one wants to talk about.
The answer to your question is clearly not. In fact, no one really knows where it comes from. To the fifth circuit, the government argued that the definition of the term "Unauthorized alien" in 8 USC 1324a(h)(3) gave it such authority. To the Supreme Court, the government argued that Texas was wrong to focus on 1324a(h)(3) and that it was really 8 USC 1103(a) that gave the authority.
If you read either of those provisions, you can see how silly the claim is.
That is reenforced by the legislative history.
One of the amicus briefs filed in Texas addressed this employment question.
The D.C. Circuit is scheduled to hear arguments on the specific issue of work authorizations next month.