Perhaps so, but crossing the border without an entry visa is more often than not a misdemeanor offense.
The kind you're talking about -- aren't felony criminals. They crossed the border illegally --- if they didn't obtain stolen SS cards by hooking up with organized crime rings which provide those, and they aren't committing other crimes --- then you're right. But these are going to be denied Social Security checks unless they will lie and claim they really did steal a Social Security number.