The only time I heard such an argument, it wasn't their true rejection, and they invented several other such false rejections during the course of our discussion. So that would be my response on hearing someone actually make the unaddressed argument you outlined.

Do such game-theoretic reasons actually hold together, by the way? It seems unlikely, unless you suddenly start caring about children somewhere during their first few years.

No, this is definitely my true rejection. To expand a bit, take the infanticide case as an example: I think infanticide should be illegal, but I don't think it should be considered murder or anything close to it, nor punished nearly as severely.

Basically, there's no "real" line between sapience and non-sapience, and humans, in the course of their development, start out as cognitively inert matter and end up as sapient beings. But since we don't think evaluating in every single case is feasible, or reliable in the "border region" cases, or likely to lead to consistently (morally) good outcomes in practice (due to assorted cognitive and institutional limitations), we want to draw the line way back in the development process, where we're sure there's no sapience and killing the developing human is morally ok. Where specifically? Well, since this is a pragmatic and not a moral consideration, there is no unique morally ordained line placement, but there is a natural "bright line": birth. Birth is more or less in the desired region of time, so that's where we draw it.

Now, since we drew the line for pragmatic reasons, we are perfectly aware that the person who commits infanticide has not really done anything morally wrong. But on the other hand, we want to discourage people from redrawing the line on an individual basis, from "taking line placement into their own hands", so to speak, because then we're back to the "evaluating in every case is not a good idea" issue. But on the third hand, such discouragement should not take the form of putting the poor person in jail for murder! The problem is not that important; the well-being and happiness of an adult human for a large chunk of their life is worth more than the (nonzero, but small) chance that line degradation will lead to bad outcomes! Make it a lesser offense, and you've more or less got the best of both worlds. (Equivalent to assault, perhaps? I don't know, this is a practical question, and best settled with the help of experts in criminal justice and public policy.)