First off, the NAP is based upon private property, not the other way around.
Second, anarcho-capitalism is a relatively new ideology and you cannot hold all past proponents of it as the God of capitalism. It has been developing over the past decades. I would say most ancaps these days have disregarded the notion that you can let your child die. We believe that if you bring a baby into this world that you have an obligation to see that the baby is cared for and not left to starve.
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"The NAP is based upon private property, not the other way around."
That's my entire point. Private property is based upon aggression, so the institution of private property conflicts with the NAP. There is an inconsistency in the AnCap position.
I also recognize that most modern AnCaps don't agree with Rothbard's defense of infanticide. However, that is besides the point because Rothbard's position clearly follows from the AnCap position, so modern AnCaps who disagree with Rothbard on this point are not being consistent with their own principles.