The claim that the passage in Leviticus 12 makes childbirth "a sin" is patently ridiculous. What is lacking here is an understanding of the concept of "clean and unclean" in the Hebrew ritualistic system. The idea of being "unclean" denoted ritual uncleanness, and was a typological representation of sinfulness, but did NOT necessarily imply actual sin on the part of the "unclean" person. The whole concept is a didactic picture designed to illustrate certain spiritual truths to the Israelites. One of these truths, for instance, is God's separation from spiritual death, which is typified for us in that those individuals who had touched dead carcasses (either animal or human) were considered unclean for set periods of time. For instance, in Leviticus 11:28, anyone who moves a dead carcass was unclean until the evening and after they had washed, likewise with one who eats from a carcass (either the meat or something growing from a carcass) was also considered ceremonially unclean until the evening after they had washed. The actions of touching a carcass, etc. didn't mean that a person HAD SINNED. It was merely a picture designed to teach the Israelites that, just as touching a carcass made them ceremonially unclean for a time, so does allowing yourself to interact with sin (which death represents) cause you to become polluted in your own life. Uncleanness was a typological picture of the stain of sin, but wasn't itself sin.
That being ceremonially unclean did not necessarily imply sin on the part of the unclean person, we see the following example in Leviticus chapter 16. This chapter deals with the Day of Atonement, a ceremony in Israel in which atonement by sacrifice was made for the nation. In verse 27, we see that the carcass of the bullock used for the sin offering of atonement was to be taken outside the camp, including its dung, flesh, and hide, and burned. This was a direct commandment from God concerning the disposal of the atoning bullock. In verse 28, we see then that the person who went out to burn the carcass was afterward to wash his clothing and his flesh, and then enter the camp. This was because this person was ceremonially unclean until they washed, just as one who touched a carcass was in Leviticus 11. Did this person sin to make themselves unclean? Of course not, they were in fact carrying out the Lord's command. Their uncleanness was of a ceremonial nature, not the result of a sin on their part. It served instead to remind that person that even though they were doing the Lord's work, it is still possible for them to become spiritually unclean in their lives through sin.
Thus, we see that the claim that the purification required for a woman after childbirth was the result of some "sin" of childbirth is not consistent with the typology presented in the Biblical pattern of ceremonial cleanliness presented to Israel in the Pentateuch. The reason a woman was ceremonially considered unclean after childbirth was because, through childbirth, another person has been brought into the world with an inherent tendency to sin and who is by nature separated from God. In other words, a spiritually dead person has been brought forth, and a time of ritual purification was required to illustrate to Israel the seriousness of sin and the spiritual death which sin has brought on mankind. As with the example of the person who took the carcass of the atonement bullock outside the camp to be burned, a woman who gave birth to a child was indeed fulfilling the commandment of God to "be fruitful, and multiply" (Genesis 1:28), yet also had to endure the period of ritual uncleanness, but NOT through any "sin" on her part.