The krias Hatorah of the Cohen Gadol on Yom Kippur is an integral part of the Seder Avoda. Rabbi Yochanan says that is required, because Moshe Rabbeinu read the parsha of miluim.
If so, why does the Cohen Gadol read the parsha after he has basically finished the avoda, but Moshe Rabbeinu read the parsha before anyone started doing anything?
Moshe Perkal
Rashi on 5b learns that Rebbi Yochanan is refering to the Milu'im, not Yom Kipur. However Rashi on 68b does say that we learn Yom Kipur from Milu'im and implies that the Kri'ah is m'Akeiv as you write. The Me'iri brings two opinions as to whether the Kri'ah is m'Akeiv.
As to why it was left until so late in the day, see Tosfos Yeshanim 68b who explains that this juncture is appropriate since there was anyway a break in the Avodah (whilst the Sa'ir was taken to Azazel and Par and Sa'ir were burned). Tosfos ha'Rash suggests that the Kri'ah was left until after the Avodah because the Avodah is Mechapeir. This is a good answer even according to those who hold that the Kri'ah is m'Akeiv.
Furthermore, we know that time was at a premium on Yom Kipur, so we could say that this was an oppurtune time since they had to wait for the Sereifah of the Par and Sa'ir which did not require the Kohen Gadol, so no time was taken up with the Kri'ah.
Dov Freedman
Thanks you for the quick and interesting answer. I would like to add that the Tosfos Yeshanim on 68b you pointed out also quotes a Yerushalmi which brings a posuk to explain why the Kriah was after the Avodah. This would also explain why the kriah was not done during the longest break of the day,waiting for the sa'ir to reach and desert.
Moshe Perkal
The Yerushalmi brings the Pasuk as a source for the Keri'ah, not the order. See Chidushei Maran Riz ha'Levi (page 77, column 1, first paragraph) who offers a novel explanation as to why the Keri'ah could not be done earlier, in the name of his father, the Grach.
Dov Freedman