The Mishna Yoma (68b) says that the Kohan Gadol would be informed when the שעיר עזעזל reached the midbar and then he would read the Torah. Why the need to know the שעיר already reached the midbar? Rashi says based upon pessukim that the Kohan Gadol can't start another avodah until the previous one is completed. The Sfas Emes challenges Rashi and assumes it is just a Rabbinic law to ensure one mitzvah is done before completing another. The way the Mahar"i Korkus understands the Rambam, this is not a general law that the Kohan must wait before starting another avodah but this is a unique law for the reading of the Torah that the Kohan must wait until the preceding avodah is done. However, this is very hard to understand why would the reading of the Torah be different than other avodah? The Briskors (Rav first letter in back of sefer, Grac"h on Shas, Harraray Kedem #77) say that the Rambam holds there is a general law for the kohan to remain in the place of the avodah until it is completed. Therefore, the Kohan must remain in his place before moving to the Ezras Nashim to read the Torah. It is hard to hard why there would be such a law and why if the שיער it sent out anyway can't the Kohan Gadol leave?
The possuk Rashi cites is שנאמר ושלח את השעיר במדבר ואחר כך ואת חלב החטאת יקטיר המזבחה. How does Rashi derive from here that reading the Torah which is Rabbinic and not an avodah must be delayed as well? The simple answer is that Rashi will reject the assumption of the question.
Rashi may indeed hold that the reading of the Kohan Gadol is the a Torah law that is considered bound by the laws of the avodah. In the very next Mishna, in the opening of the next perek, Rashi says that the reading of the Kohan Gadol is derived from the מילואים. The Sfas Emes disagrees and assumes it is a Rabbinic law. However, Rashi is clear it is a Torah obligation.
The Mishna says that the Kohan Gadol may read the Torah either in a white cloak of his own or the בגדי כהונה. The Gemarah refers to the reading not as an avodah itself but as a צורך עבודה. The Rambam in Perush HaMishna says it is not an avodah and the proof is that it does not require בגדי כהונה. The Netziv says that Rashi holds since the reading is considered צורך עבודה it also needs the previous step to be finished. The Mishna says there are 8 berachot recited by the Kohan Gadol. Rashi says all the berachot are recited after the reading but the Rambam holds that the regular berachot before reading the Torah are recited. The Gevorot Ari wonders why Rashi doesn't hold that there is a beracha beforehand like any other Torah reading? However, if this is a reading not done as a chovas hayom like other reading but is viewed as part of the avodah service and hence it does not have the rules of regular reading.