The Christian New Testament sheds some light.
The sin which Adam committed in the Garden of Eden had resulted in his separation from God's life. That is, Adam was condemned to death (Gen 2:16-17). His immediate separation from God resulted in spiritual death, i.e., his access to the tree of (eternal) life was terminated according to Genesis 3:22-23. So Adam's spiritual death eventuated in his physical death: thus we read, ". . . ashes to ashes, dust to dust" (Gen 3:19).
When human beings are born, this spiritual death is transmitted to each human being, and thus each human is born spiritually dead (Rom 5:12). Of course, like what happened to Adam, all human beings eventuate in physical death also.
In the Hebrew Bible (Old Testament) death is "dirty." Thus dead bodies (whether animal or human) are "dirty." Any creatures that thrive on dead waste --for example scavengers-- are also "dirty."
So when human beings are born, the transmission of spiritual death is "dirty." Therefore sex is not dirty in the Hebrew Bible, instead what is "dirty" is the spiritual death, which is transmitted from parents to children.
For example, the emission of semen (Lev 15:16-17) or even the menstruation of the female egg (Lev 15:19-24) are "dirty" not because they are functions of the body fluids of sexual organs, but because spiritual death is procreated through these activities of the human body. Sex therefore is not "dirty" in the Bible. What is "dirty" is spiritual death. As noted above, spiritual death eventuates in physical death, which is "dirty" as we noted. Death is "dirty."
In the Christian New Testament, spiritual death is washed away clean with eternal life (water). This living water is available, because sins/transgressions were removed through the sacrifice for sin.
That is, the eternal life of God was incarnated in flesh, but without the transmission of spiritual death--that is, the "father" of Jesus was not a spiritually dead mortal man, but the living God. As the sacrificial lamb for sin he was therefore not "dirty" because he was NOT spiritually dead. He was born the eternal life of God incarnated in human flesh, thus as the lamb he was without spot or blemish (1 Pet 1:19) -- he was not "dirty." When his body was made to be sin, it was then that he therefore had died.
But while his body was sufficient to be judged for sins, it was his eternal life that had "abolished" spiritual death (2 Tim 1:10 in NASB) and therefore his subsequent physical resurrection had followed. That is, his body was the sacrifice for sin, but his eternal life was at one and the same time "indestructible" (Heb 7:16 in NASB) -- it was therefore "impossible" for death to hold him (Acts 2:24). So the sinner, whose sins were judged through the body of Jesus, could also receive the "washing" of the living water of eternal life through him. The birth of spiritual life through him is thus termed to be "born again" (Jn 3:3-7 and 1 Pet 1:3).
This birth however is not "dirty" like the birthing of the flesh, but is clean because the birthing is eternal life through the Spirit of God, who removes spiritual death with the water of eternal life (Titus 3:5). This baptism (washing) in eternal life removes the spiritual death of Adam.
As a closing observation, when seminal emissions occurred, or when menstruation occurred (unrelated to any birth), then interestingly enough it was only "water" which was the means of cleansing (cf. Lev 15:16-17 and Lev 15:19-24, respectively). But when an actual birth occurred with the woman, there was an offering for sin (Lev 12:1-8), because the "sin" is Adam's disobedience, which creates spiritual death in the newborn baby. The condemnation of Adam's sin (spiritual death) is therefore transmitted to each and every human being (Rom 5:12). Jesus died to take away sins and transgressions, and in turn, to provide eternal life, which removes the spiritual death.
Thus the Christian New Testament sheds light why a sin sacrifice was required after the birth of a child in Leviticus 12:1-8.