Why are the deaths of Nadav and Avihu recounted on Yom Kippur?
The Talmud teaches the general principle: The death of the righteous atones for sin:
אמר רבי אמי: למה נסמכה מיתת מרים לפרשת פרה אדומה לומר לך מה פרה אדומה מכפרת אף מותתן של צדיקים מכפרת
“Rabbi Ami said: Why is the death of Miriam next to [the laws of] the Parah Adumah [*]? To tell you that just as the Parah Adumah atones [**], so too, the death of the righteous atones.” 
אמר רבי אלעזר: למה נסמכה מיתת אהרן לבגדי כהונה מה בגדי כהונה הכפרין אף מיתתן של צדיקים מכפרת
“Rabbi Elazar said: Why is [the death of] Aharon [next to laws of] priests’ clothing? [To tell you that] just as priests’ clothing atones, so too, the death of the righteous atones.” 
In parshah “Acharei Mot,” (Vayikra/Lev. 16), mention of the deaths of Nadav and Avihu is immediately followed by the account of the atonement procedure using the “scapegoat.” The same principle as above would seem to apply, for the same reason.
But “vicarious atonement” is denied by Yehezkiel/Ezekiel: “The soul that sins shall die. The son shall not bear the sin of the father, nor shall the father bear the sin of the son: the righteousness of the righteous shall be upon him, and the wickedness of the wicked shall be upon him.” 
The Talmud, however, doesn’t seem to be saying that the “righteous” die because of our sins; rather, their death is a loss to us, and thereby atones for us
The Zohar explains why the selection from “Acharei Mot” is recited on Yom Kippur:
וכל מאן דמצטער על יסוריהון דצדיקיא מעבירין חובייא דלהון מעלמא
ועייד ביומא דא קורין אחרי מות שני בני אהרן דישמעון עמא ואצטערון על אבודהון דצדיקיא ויתכפר להון חובייהו
וכל דמצטער על אבודהון דצדיקיא או אחית דמעין עלייהו קדשא בריך הוא מכריז עליה ואומר וסר עונך וחאתך תכופר
“All who have distress for the suffering of the just will have their sins pass from this world.
“For this reason, on Yom Kippur, we recite [perek “Acharei Mot”] “After the death of the two sons of Aaron,” so that people will hear and feel distress for the loss of the just, and their sins will be forgiven.”
“For all who feel sorrow for the loss of the just, or who shed tears for them, the Holy One, Bless Him, announces, ‘Your iniquity is taken away; your sin is purged’ .” 
Note: The Zohar doesn’t mandate this reading for Yom Kippur. Rather, it comments on why it was in place – i.e. it was already customary when the Zohar was taught. It should also be noted that Nadav and Avihu don’t die in this passage; the passage begins “After the deaths…”
The association between the Zohar’s teaching and the Yom Kippur parshah isn’t “arcane.” It’s mentioned in many popular, accessible sources.
But if we take the Zohar literally, it’s not Nadav’s and Avihu’s deaths that atone, nor simply the reading of it alone. Only those who become “upset” by reading/ hearing about their deaths, are given atonement. It’s our reaction to their deaths that brings us atonement; not their deaths alone, as in the Talmud [although perhaps that’s what the Talmud is implying].
It’s as if the Talmud is saying: The deaths of the righteous are a loss to us all. To which the Zohar is adding: Especially if we’re selfless enough to realize it!
Perhaps the reading of “Eleh Ezkrah” — “The [Deaths of the] Ten Martyrs” — is included for a similar reason. It might even give added meaning to the reading of “U’n’taneh Tokef,” especially when introduced by mention of Rabbi Amnon’s death.
I’ve found myself having more empathy for Aharon, who experiences worse pain at losing his sons, than for Nadav and Avihu, who didn’t seem to experience suffering in their deaths.
For that matter – in the course of the latter part of Torah, Mosheh experiences the deaths of his brother, sister, nephews and (presumably) his wife. Isn’t he suffering, too? Shouldn’t we feel some rachmanut for him? Is it atoning, or at least purifying, for us to do so?
Who are the “just” or “righteous?” Those who are innocent of sin. It doesn’t mean that they never sinned; it means that they’ve repented of their sins, to the greatest extent possible.
What if we’re troubled by the sufferings of those who die – innocently – in natural or man-made disasters (not of their own doing)? Even if we believe that this is their “karma,” aren’t we purified to some degree – in our hearts, at least – when we’re upset by news of tragedies – especially to children? Like the death of a child at the hands of an abusive or neglectful parent; the deaths of children in pervasive famines (as in Africa, now), or in abusive, dangerous (child-)labor practices (especially where otherwise illegal, too)?
It also seems to me that the narrative of the deaths of Nadav and Avihu, and Aharon’s quiet acceptance, both echo the theme of the Akeida, which is read on Rosh Ha-Shanah. Especially for the m’kubalim in Sfad, there was only one day of RH and one day of YK; thus the two Torah readings could be thematically “bookended.”
Regarding cantillation (the melody of the public Torah-reading) on Yom Kippur, only Ashkenazim, among all Jewish traditions, use a special “nusach” or musical mode for this Torah-reading. Others use the same mode as for any Torah-reading during the year. Because of the Zohar’s statement regarding the atoning power of our tears for Nadav and Avihu’s deaths, it became the Ashkenazic custom to read the Yom Kippur-parshah in the mode used by Sephardim for reading “Job” on Tisha b’Av  – a mode that’s described as expressing “complaint and sadness.”  Its use was later extended to the Rosh Ha-Shanah readings as well. The Minchah (afternoon) reading for Yom Kippur reverts back to the standard Torah mode.
(The High Holiday Torah reading can be heard on www.virtualcantor.com, as well as other online sites.)