Executorship of an estate doesn't have anything to do with guardianship of a decedent's child.
The only claim he has to that child is his name on the birth certificate. At best, he gets a presumption, but presumptions can be rebutted. The fact that ANS (probably under Stern's control) did everything to avoid a paternity test tells me that the baby was sired by Birkhead.
We were talking about what might be in any will ANS might have had, not about guardianship of the baby, unless there is a will and there are provisions.