The Book of Common Prayer refers to the office of the "Churching of Women". I suspect that it's not seen as being very politically correct these days to call it thus. Of course, this is only the subtitle of the service: the BCP gives it its proper title of "The Thanksgiving of Women after Child Birth". Certainly, parts of the Church will do Thanksgivings before Baptisms and often that is the only service the child will ever receive from the Church. For some, the Thanksgiving service has become the naming ceremony and many families will understand that this is the only time they need to go to Church with their child.
For the Jewish Community, the rather bloody nature of childbirth rendered a mother ritually unclean and thus unable to take part in the fulness of Jewish rites until she had been purified. This does sound rather negative, that the woman, through no fault of her own has somehow become impure simply by giving birth to her child. Many today might read into it misogynistic overtones there, of some oppression of the woman. One might suggest they read such overtones in because they want them to be there.
Considering that a mother would actually need some time to rest and recuperate, perhaps ritual impurity could be seen as a way of regaining one's strength and healing after an arduous ordeal for mother and for baby. Some time out of society would provide this.
Of course, impurity and cleanliness were important to the Jewish way of thinking. The laws which God gave the Jews were ways in which they could preserve a distinct identity as people of God. Cleanliness meant a clear identity as one of God's Chosen People and to be unclean meant the possibility of some contamination with that which was outside. Blood was seen as a venerable substance as the life force. Considering that other tribes around the Jewish Communities were blood-cults which enjoyed the spilling of human blood as part of their rites, the rather more sparing attitude of blood was God's way of telling the Jews, "you are not like them."
One can see this when Elijah confronts the Prophets of Baal. The Prophets entreat Baal to light the fire for their sacrifice and when he does not, they cut themselves in order to entice the false divinity into action. For the Jew, blood is only something that is spilled in sacred covenants and sacrifices, not to be treated frivolously or consumed or even worshipped. It was part of the pact between Israel and God, and thus only to be treated in accordance with that pact.
Thus, even when the blood is one's own, a Jew would not wish to be identified in any way with the cults of old and thus would seek to have their part in society re-established through a cleansing rite. It was something they wanted to happen - the Jewish Community was something that each individual rejoiced in.
So now here we are. Our Lady has come to be purified at the temple. Whether she needs it or not is perhaps a matter between her and God and not for our over-curious minds, but nonetheless she seeks, as any other Jewess, to be part of the Jewish Community and for her newborn son to be part of that community too. It is a community absolutely integral to her way of life and she intends fully to bring up her son in that same beloved community.
It is the aged Simeon who has the startling revelation in his Nunc Dimittis. The Child is to break through the barriers of the Jewish Community and open it to the Gentiles too. How will He do so? Answer: through the pouring out of His blood. It is startling to think that a culture which avoided blood would find itself with the opportunity to be transformed by that very substance. This is another reason why the Lord Jesus is a stumbling block for the Jews. However, the pouring out of the Blood of Christ represents a new identity for the human soul. Not a blood cult like Baal, but a New Covenant in His Blood.
Thus the Feast of the Purification becomes a prediction for the existence of the Church, of the new identity, of the Baptism into Christ and a symbol for our own part in that community.
At her Churching, a woman brings her new baby to reaffirm her identity as Christian and to introduce her Christian family to the newest member. The next step for that little one is obviously Baptism where the child is given that new identity of its own apart from the mother. If we look down on this action then we denegrate both the offering of the woman, but also of the miraculous act of childbirth peculiar to that sex.
The example of Our Lady shows a commitment to playing a part in society, of not wanting to go one's own way, or to demand one's own path in Life separate from everyone else. She is making a decidedly Anti-Individualistic statement, seeing her identity within Holy Church and only as part of that Church. Outside of Church, she is just Mary; within it she is Theotokos - the Virgin Mother of God.
How do we show a similar commitment to the good of our society?