“If the benefits dry up they will self deport.”
They will not sign up for amnesty unless it is profitable for them to do so. Without amnesty they can work under the table, get medical care for free, and apply for benefits for themselves and their kids. If they sign up for amnesty, they have to be at least partially accountable. Why do that?
How does it profit them if they sign up for amnesty? They won’t be deported if they don’t.
Well, there’s tax credits where freeloaders who don’t pay taxes get free money.