No Leon, coorelation does not imply causation.
High birth rates and poverty go togther, true, but both are to blame. If you take the 'natural state of humanity' as hobbes did, as nasty, brutish and short. Hobbes also said poor and solitary.
With poverty and large families together we have something less solitary than Hobbes imagined. Safer in large family groups. And entirely valid.
But there is an alternative. You can choose to co-exist in the modern society which we have built, one without this kind of poverty, you can choose to accept that we thrive when we act collectively. When we pay taxes, build roads, schools and hospitals.
The natural responce to poverty is to have lots of children, at least some of them will make it. Your responce os to ignore that families were usually bigger in the past, and some people have not had the advantage of leaving that poverty stricten past behind.
You instead blame the support society has offered to help them out of that situation. Like they started having bigger families because of social welfare? Do you have any evidence? Or is it just te logic of a person who hasn't lived in poverty, how many families living off of state assistance have you shared a home with? How did you gain your insight into their decisions? Other than arm-chair logic??