If she wants the father to help support raising the child and he refuses to sign the BC, then yes. If she doesn't want him involved and he doesn't petition to sign it, she's on her own.
Are you assuming here that WIC, EBT's and every other sort of State-as-father-of-all stuff is 100% gone?
That would be nice, but is irrelevant to this discussion on The State being extracted from marriage.
It can't be based on supernatural revelation, to which non-believers or different-believers cannot be held.
It can't be based on esthetics: the "ick" factor, for other tastes, might be a "yum" factor.
It can't be based on something inherent in small-group sociology, since there have always been non-marital small-groups (business partnerships, siserhoods, quilting bees, barbershop quartets) --- but the ones that correspond to "marriage" always correspond to "man-woman."
So if you're interested in the core reason for marriage, you'll find it is necessitated only by procreative potential: the possible transmission of life to a new generation. That's why you need some structure to provide for kinship, nurturance, socialization, and property transfer.
It's this begetting-and-raising issue which is the key. No other factor is significant from a State or public-interest point of view. Without the potential for procreation, the society at large has no real interest in your amours.
So that's why WIC and EBT cards are relevant to the discussion. All government commitments to households will grow as the marriage commitment recedes. As Dad-ism is vanquished, Statism triumphs. If the husband/provider is un-attached, the bureaucrat/provider takes his place. You can either have the marital estate, or the Leviathan State.