Meaning, in a general sense that a natural born citizen can get benefits but a naturalized or legal immigrant with a green card cannot.

I am certain that is not what the bill means. Legally the term "immigrant" refers only to lawful permanent residents (green card holders), and not to visa holders or naturalized citizens. U.S. law does not make any distinction whatsoever between naturalized citizens and "natural-born" citizens; every citizen is simply a citizen. This is about as inviolable as anything in US law. There has never been any law that groups naturalized citizens together with non-citizens and separate from citizens at birth, the way you suggest.

As to the argument about tourists who are here on visas being denied social benefits? That's a stretch

Yes, it is absurd and that is my point. The right to receive public assistance (upon relevant qualification) is not something everyone has, the way freedom of speech or any other civil rights are. The equal protection clause applies only to those rights that everyone has; other privileges which are legitimately restricted to citizens or certain immigration classifications are not. If you claim that public assistance falls under equal protection, then you are saying it is among those rights that everyone has.