It's difficult to believe that the number of illegals leaving is greater than or equal to the number of those staying and increasing their population by having kids.
I don't see why you turn your back on facts as reported by a reputable source.
But you touch upon a good point - a guest worker program would allow workers to come here when needed and go home later - to their families. The tougher you enforce the immigration laws, and the fewer meaningful opportunities you provide to come here and work legally, the more you encourage the situation of illegals coming here with their families. If they can't go home and return, they'll bring their home here - and create the problems you fear for the social welfare system.
All these problems are the result of poor government policy, and, secondarily, the greed of employers. As such, the folks I blame the least are the illegals themselves. All they want to do is provide an honest day's work for the sake of their families.