Your reading is more along the lines: "Breastfeeding - when and where it makes sense to do so - is a necessary part of taking care of a newborn that men cannot do."?

Isn't that like "Wearing a bicycle helmet - when and where it makes sense to do so - is a necessary part of riding a bicycle."

That is, your addition negates the seemingly firm statement of "necessary", while the later use of "vital" seems to support the firm intent of "necessary".