In his commentary on Leviticus 4:2, where the Torah introduces the קרבן חטאת ("sin offering"), R' Samson Raphael Hirsch explains its purpose with:
The offering, with which a soul that has fallen out of focus of the Will of God which should form the centre which directs all its actions, seeks to regain the nearness of God, is called a קרבן חטאת.
The essential aspect is the return from falling out of focus, not "sin," per se. You can see this from the fact that the Red Heifer is also called a "חטאת" (e.g. in Numbers 19:9), despite the fact that it is used to recover from טומאה (moral depression, according to R' Hirsch) from contact with death, not from sin.
According to R' Hirsch, טומאה is a mental condition that would prevent a person from participating in holy practices with the correct mindset. In particular, many types of טומאה cause this problem by afflicting a person with the illusion that people are trapped by the physical world and have no genuine free will. One can't participate in the holy service, in which one dedicates some aspect of oneself to God, without a complete sense that one is approaching God out of free choice.
Similarly, according to R' Hirsch in his commentary on Leviticus 12:6, the חטאת brought by the new mother helps her refocus following טומאה, not sin. As he explains in his commentary on the previous verse, in the process of giving birth, a woman necessarily surrenders to the overwhelming physical process and thus is intimately subject to the illusion that she is an unfree object of natural forces rather than the holy, volitional being that she is. So, on 12:6, he explains:
That is why she first vows in her חטאת העוף, that she will not allow her power to determine her own actions in all moral matters to be broken by the painful days which are inseparable from her high calling in life; but rather, by undertaking and enduring such days of suffering in the spirit of duty, for the sake of fulfilling her high mission in life, she exercises her power of free will even in this, and so transforms the passiveness of pain itself into active moral energy.
Isaac Levy translation