It was if you want to use family as total support for welfare applications and not just the individual, which is what you were arguing BTW whether you knew it or not. The children of illegals are US citizens and are entitled to all rights and privileges therein. Thus, the US citizen child can have welfare benefits because the child has no assets and very little family support. Whether or not the parents are US Citizens, Legal immigrants or even illegal immigrants really does not matter.
But you were stereotyping when you said The part I bolded is the generalization. The similarities about black people is also a common theme among certain segments of society as well. May want to look at the issue of White Rock Chapel in DFW area as another prime example of the attitudes of those living affluently.
I'm not sure that I regard my statement, "Illegals are poor people " as a stereotype, anymore that saying that people with section 8 vouchers are poor; it's simply an observable economic result of their condition. It's extremely difficult to be rich or middle class and illegal because all legitimate ways of earning wealth are cut off from you.