There is pretty much exactly one thing that only a mother can do once the baby has been born, and you just named it.
For many reasons, a mother might not be willing or able to breastfeed her child or to do so exclusively, so you'd need additional feeding arrangements in those situations.
Even if the mother is breastfeeding, in practice she might appreciate being able to express some of her milk and have the father feed the baby sometimes.
Things could be much more even with early years childcare than they are in many places at the moment, and in my experience there are a lot of mothers and a lot of fathers who wish it were so.
For many reasons, a mother might not be willing or able to breastfeed her child or to do so exclusively, so you'd need additional feeding arrangements in those situations.
Even if the mother is breastfeeding, in practice she might appreciate being able to express some of her milk and have the father feed the baby sometimes.
Things could be much more even with early years childcare than they are in many places at the moment, and in my experience there are a lot of mothers and a lot of fathers who wish it were so.