SS benefits are not a current issue because there is no surviving child.
CA claims she had papers drawn up, not by "Paul" but by someone to whom they were referred by "Paul." I don't remember the discussion from that time well enough to recall if it was determined to whom she referred or how this was debunked at the time.
See CA's discussion of this starting around 3:00 here.
CA also claims in this video that she was told the father was "Eric" or one other person. She states she never heard of the "Ortiz" family. I would guess that TANF or other welfare type benefits were not available because nowadays, one is required to name the other parent so that the state may try to recoup some of the costs through child support. I don't think KC would or that CA would allow her to claim "unknown" or how the welfare department would handle such a claim. I've read cases where the state paid for multiple DNA tests to determine a father for a child when the mother kept making wrong guesses when applying for benefits. I don't think either KC or especially CA wanted to admit they had no idea who fathered Caylee or to go through maybe a dozen guys without a match.
I think the JG debacle was enough of a lesson for them on this. Seemed like they were interested in getting some legitimacy and some child support enough at that time to deal with the "other" family. (I also suspect CA knew all along that JG was being scammed.) No one, especially CA, wanted to face the humiliation of another exclusion through DNA testing.
Discovering the identity of a viable father today may prevent some future profits if the father were to gain control of Caylee's "estate."