It's just an arbitrary moment in the child's development.
That kind of talk is crazy talk, irrespective of your position on abortion. There are a number of hugely significant and factually obvious differences that divide the period prior to birth and the period after. Feeding is just one example that comes to mind within 10 seconds of thinking about it.
To say "the moment of birth is not special with respect to personhood" is to reveal a tacit conclusion about personhood: that your personhood predicate is not relational but solely based on intrinsic properties of the organism.
Empirically, it turns out that many of the nodes associated with the node "personhood" hinge on relational factors: a baby who has no one to talk to will not develop language, for instance (the "critical period hypothesis").
Similarly to say that the moment of birth is not special, with respect to how one should view the act of killing a baby, is to reveal a tacit assumption about what makes killing a baby "right" or "wrong": that only the difference it makes to the baby's future prospects should be taken into account.
Empirically, a death before or after the moment of birth makes a huge amount of difference as far as the mother's interests are concerned, for the same type of reasons that make the moment of birth special: feeding and other caretaking behaviours kick in at that moment. (Granted, some forms of bonding start well before birth. The psychological aspects of pregnancy are complex, as the phenomenon of pregnancy denial illustrates strikingly.)
My intuition is that (as social animals) there is much more to our attitudes toward babies' deaths than our unconscious evaluations of their reproductive potential, and the conclusion that "if our attitudes were allowed to drift toward a new evolutionary equilibrium which took account of technology, we'd come to hate abortions again" strikes me as seriously unfounded.
I don't understand your third paragraph. The moment of birth doesn't affect the unborn baby's chances of having someone to talk to, conditioned on survival.
And I don't understand your last paragraph. Try this thought experiment: how would you design a human whose only goal were to have more kids in the modern world? What would be its attitude toward abortion? Or you could look at groups of humans who reproduce faster than average today, like the Amish or Orthodox Jews, and their attitudes toward abortion. They are probably a good indicator of the evolutionary equilibrium we would converge on.
Some thoughts that I don't remember anyone expressing on LW.
First let's get this out of the way: life does not "begin at birth". As far as life can be said to "begin" anywhen, it begins at conception. Moreover, the child's intellectual abilities, self-awareness or similar qualities don't undergo any abrupt change at birth. It's just an arbitrary moment in the child's development. So it would seem that allowing killing kids only before they're born is illogical. What are the odds that your threshold for "personhood" coincides so well with the moment of birth? Could it be okay to kill kids up to 2 years old, say? CronoDAS voices this opinion here.
But there's another argument in favor of considering the moment of birth "special". Eliezer linked to a study showing that the degree of parental grief over a child's death, when plotted against the child's age, follows the same curve as the child's reproductive potential plotted against age. Now, the reproductive potential of an unborn kid depends on its chance of survival, and the moment of birth is special in this respect. In the ancestral environment many kids used to die at birth. And mothers died often too, which made their kids less likely to survive. An unborn kid is a creature that hasn't yet passed this big and sharply defined hurdle, so we instinctively discount our sympathy for its reproductive potential by a large factor without knowing why.
How much this should influence our modern attitudes toward abortion, if at all, is another question entirely. As medicine becomes better, kids and mothers become more likely to survive. So if our attitudes were allowed to drift toward a new evolutionary equilibrium which took account of technology, we'd come to hate abortions again (thx Morendil). But then again, the new evolutionary equilibrium is probably a very nasty system of values that no one in their right mind would embrace now (won't spell it out, use your imagination).
Ultimately your morality is up to you and the little voices in your head. You think womens' rights trump kids' rights or the other way round, okay. But if you use factual arguments, try to make sure they are correct.
ETA: see DanArmak's and Sniffnoy's comments for simpler explanations. Taken together, they sound more convincing to me than my own idea.