What makes you think that unmarried mothers are AUTOMATICALLY more competant to care for a child than their fathers are? Just because a Dad is unmarried does NOT make him necessarilly any less capable a caregiver than you are.
As stated Dad will gtet his OWN visitaion time with his child, if he files, fairly quickly. Breastfeeding is not an excuse to deny a dad their own parenting time. That's what breast pumps are for.
The ONLY caregiving duty a dad cannot perform is lactation. The guys I know did everything else for their babies from birth.