Empirically, families used to stay together and now they don't. The 'why' is very sticky and gets into things like obscure divorce law changes, the rise of feminism, etc.
I'm not sure that it has anything to do with the social safety net; rather, I should say the causality is backward. We need the social safety net because the nuclear family is on the decline. Reforming welfare may not do anything about that. (In fact, it could easily do the opposite -- cf welfare babies.)
From my point of view it seems an insoluble problem, I confess.