Well, at the risk of potentially coming off wrong, I’ll say that there certainly is room to make this argument assuming you come from a viewpoint that abortion should be fine anytime up until birth. If you don’t agree with that view, then any further argument on this point is probably moot since it is ruled out automatically (Except if you want to make exceptions for disabilities discovered upon birth).
But if you believe that abortion is moral up until birth, than its not really as big of a stretch as it might seem to say that using birth as a bright line is wrong. The simple act of being born doesn’t make the newborn somehow far more developed than it was the moment before. Of course, one major benefit of using birth as an absolute boundary is that it prevents the issue of determining at which point after birth it stops being OK to terminate.
Not having read the piece, I cannot really comment further on it, although I will admit it doesn’t sit all that well with me at first blush.