Acharei Mot details the special avodah, sacrificial rites, that the High Priest would perform on Yom Kippur to effect atonement for the Jewish People. However, as the Vilna Gaon in Kol Eliyahu already noted, the Torah only introduces the connection to Yom Kippur at the very end of the lengthy description of this special avodah. The framing of the avodah is not what must be done to achieve atonement on Yom Kippur but rather what must be done when Aaron wants to enter the inner sanctum:
And the Lord spoke to Moses after the death of the two sons of Aaron, when they came near the Lord, and died.
Speak to Aaron your brother, that he come not at all times into the holy place inside the veil before the covering, which is upon the ark; that he die not; for I will appear in the cloud upon the covering. Thus shall Aaron come into the holy place: with a young bull for a sin offering, and a ram for a burnt offering. (Vayikra 16:1-3)
Thus, says the Vilna Gaon, this is a rite that the High Priest - or, according to the Gaon, specifically Aharon, could do at any time that he would want to enter the Holy of Holies, as well as regular rite once a year on Yom Kippur. If Aharon wants to draw near, and not die - as his sons had - then he can do this, but only if a very precise ritual is followed. Understood this way, the parsha is underscoring the dangers of unbridled religious passion, of approaching God without due care and caution, and is giving a very structured way that one - the High Priest - can channel his desire for intense, intimate connection.
This approach, however, makes the avodah a tool for the High Priest's realization of his religious yearnings, but does not address larger communal issues. This is certainly not the simple sense of the Torah, which mandates communal sacrifices for this avodah and which declares that this avodah will cleanse the Mikdash and atone for the People. It seems, rather, that while the emphasis of the avodah is not on Yom Kippur, it is also not on the High Priest entering the Holy of Holies. Yes, he must enter it, but that is a means, not an ends. What is the ends? The Torah tells us in the climactic verses declared after the High Priest exits the inner sanctum:
And he shall make an atonement, vi'kiper, for the holy place, from the uncleanness of the people of Israel, and from their transgressions in all their sins; and so shall he do for the Tent of Meeting, that remains among them in the midst of their uncleanness. And there shall be no man in the Tent of Meeting when he goes in to make an atonement, li'khaper, in the holy place, until he comes out, and have made an atonement for himself, and for his household, and for all the congregation of Israel. (Vayikra 16:16-17).
The goal is not the entering itself, not the religious experience for its own right. Neither is the goal primarily for bringing atonement and forgiveness for the Children of Israel - this occurs, but is secondary. Rather, the goal is atoning for the Sanctuary. Well, not atoning exactly, for what atonement does the Sanctuary need. The term used here is kaper. Now, this term is usually translated as "to achieve atonement," but in fact it more precisely means cleansing, not atoning. The goal of the avodah is to cleanse the Sanctuary from the defilement that it has endured as a result of the sins of Israel. For sins, according to the Torah, create a type of tumah. Sin defiles, both the person who performs it, and the person's surroundings. And how much more so does it defile the Sanctuary, the place of the Presence of God.
Thus, to cleanse the Sanctuary, and to cleanse the People, this avodah must be performed. And the central sacrifices of this avodah are chataot, generally translated as "sin-offerings," but more accurately translated as "cleansing sacrifices." [This is why certain tamei people, such as a woman who has given childbirth, must bring a chatat. Not because she has sinned, but because the chatat achieves a cleansing of tumah. See Sotah 15a.] The focus is not on the sin itself, but on its impact, on its defilement, and the sin-offerings, or rather, the cleansing-offerings, restore the world to as it was before, restore the person to how she was before this sin had affected her, and to restore God's Sanctuary to how it was before, so that God's Presence could continue to dwell among the People.
Now, it is worth asking how this cleansing is achieved, or can be effective. Isn't tumah the antithesis of the Sanctuary? The whole halakhic significance of tumah is that it prevents engagement with things that are holy - eating sacrifices and entering the Temple. And because tumah is the antithesis of kedusha, one of the primary responsibilities of the Kohanim and certainly, following parashat Korach, the Leviim, was to protect the Mikdash from tumah (see Bamidbar 18). And when, in the opening of Bamidbar, the camps are set up around the Mishkan, the Torah commands the People to send outside of the camp all those who are tamei (Bamidbar 5:1-4), and emphasizes the need to keep tumah away from the Divine presence (verse 3, and see also Bamidbar 35:34): "And they shall not impurify their camps which I dwell in their midst."
Why, then, does the tumah not drive God's Presence out of the Sanctuary? The question is sharpened further when we realize that of all the invalidities that can occur to sacrifices, tumah is the one problem that can most be tolerated. The Talmud (Menachot 25a) teaches that the tzitz that the High Priest wore on his forehead allowed sacrifices that were tamei to be acceptable after the fact. And, as we know regarding the korban Pesach, if most of the community is tamei a fixed-time obligatory sacrifice may be brought despite the tumah: tumah hutra bi'tzibbur. Why is it that of all problems, specifically tumah, the very antithesis of kedusha, is what can at times be tolerated?
The answer, I believe, relates to the very nature of the Temple, of God choosing to make God's Presence dwell among the People of Israel. One the one hand, tumah is the antithesis of kedusha, and having a Mikdash in our midst creates a heavy demand that we do everything in our ability to keep tumah at bay. The problem is, that because we are not God, because we are human, tumah is an inevitable part of our lives. This is certainly true terms of the ritual tumah that has been the focus of Vayikra - animals die, people die, women give birth to children, women menstruate, men have seminal emissions - such tumah is encountered every day. But perhaps more significantly, it is also true about the tumah that is a result of sin. To be human is to sin. No matter how valiant our attempts otherwise, to be human is to produce tumah.
So if tumah and sin are an inevitable consequence of our human existence, how can God continue to dwell among us? The answer to this is that God has allowed it to be so. We must do all we can to keep tumah at bay, but even when we don't, God continues to dwell among us. This is what is both acknowledged and addressed by the Yom Kippur avodah. God has given us this day not only to allow us to be forgiven and to start fresh, but also to cleanse the Temple and to allow God Godself to continue to dwell among us. And hence, this verse of cleansing the Temple ends with an acknowledgement of the inevitability of tumah:
"And so he shall do to the Tent of Meeting that dwells in their midst, in the midst of their impurity."
Of all the verses that speak about God dwelling (shakhen) among the Children of Israel, this is the only verse that states not that as a result tumah must be kept at a distance, but rather that as a result, despite our best efforts, tumah will always be present to some degree. And this acknowledgement comes exactly in the section of the Torah that speaks to how it can be tolerated - because God has agreed to tolerate it, God has accepted our humanity, and, to make the tumah manageable, God has given us a rite to cleanse the Temple and start over each year.
This, then, is why of all things tumah can be overridden. Tumah is the one invalidity of sacrifices that is both the prime invalidity, but also the inevitable one. It is the invalidity that God has agreed to tolerate, because it is part of being human, and thus the necessary cost of the Divine dwelling among human beings - "which dwells among them, in the midst of their impurity." When the tumah is communal and inescapable, it can be bracketed and pushed aside - tumah hutra bi'tzibbur. And even when the tumah occurs at a local, individual level, and may have been escapable, once the korban is offered, the korban will be accepted. We are not allowed to offer such a korban - we must avoid tumah, but if we do offer it, God is prepared to tolerate tumah once it is a done deed.
Of course, we cannot allow this Divine tolerance to undermine our awareness of God's presence. If tumah becomes too much of the norm, then the place will no longer be one of kedusha. This is why it is the tzitz that allows the tumah to be tolerated. The tzitz, with the words kodesh la'Hashem, Holy to God, worn on the forehead of the Kohen Gadol, tamid, continually, is a symbol of the continual consciousness of the Divine Presence. If in the presence of tumah the consciousness of the Divine Presence remains firm, then the tumah will be tolerated.
And thus the avodah of Yom Kippur. It is exactly the Kohen Gadol who can effect the necessary cleansing. The Kohen Gadol, who symbolizes the constant awareness of God's Presence, does the avodah rites without wearing the tzitz, because such a reminder is not necessary. The Kohen Gadol enters into the Holy of Holies, is not only reminded of God, but directly in contact with the Divine Presence. It is this connection to God, achieved through constant mindfulness and awareness, which reaches its apex on Yom Kippur. It is this connection to God that allows tumah not to undermine God's presence, but to be tolerated and cleansed. "With this Aharon may enter the holy place," he may concretize the connection to God, so that the Temple and the People may be cleansed.
Tumah in its essence it is the very thing that distances us from God, but if we work to keep God in the forefront of our consciousness, to have kodesh la'Hashem inscribed on our forehead, then this tumah will be tolerated, and God will be close to us despite our tumah. God, Who dwells among with them, despite their impurity.