Jason, I think your question is an interesting one, especially in terms of Objectivist ethics--thanks for asking it. I wonder if it can be asked more broadly, though? Not just does family require a notion of 'duty,' but does anything? There may be no innate moral obligation to do anything beyond keeping ourselves alive, if in the Randian sense, 'individual' life=value. But things suffer when you don't tend to them, and some quality-of-life is perhaps just as important as quantity-of-life. So could 'duty' just be a more deterministic way of looking at 'responsibility?' For example, I'm not sure if it is 'duty,' but I want to take care of my mom when she gets older, if I have to and I am able, because she took care of me when I couldn't-- this feels more like a trade or responsibility to me than something I HAVE to do-- I just WANT to it, to return the favor.
And just as a family may require certain things be done that any member may or may not want to do at a particular time, so do most accomplishments or productive achievements. Productive work is often not easy, often requiring hard work and hard decisions to get to a desired result. So you could ask why anyone would want to accomplish anything if it is so difficult? Because 'productive' work should enhance the quality of our lives, right? I have no kids, but I do understand as a teacher the benefits of being around (most of) them, if you enjoy seeing children grow, just like an any idea, into fruition. I've learned a lot from students, gained new perspectives, laughed a lot. So I think raising a family can be 'productive' work, despite the costs and risks involved, and the fact that children don't always grow up the way we expect they should (which can still be beneficial). Of course it's not for everyone, me included, but I'm confused-- do you not see how a 'family' (not necessarily genetic) can enhance people's quality-(and/or quantity) of-life?
