The central focus of the service was to achieve Atonement for all of Israel, and any part of the service that has to do with purification of the Mikdash is just a means to the ends of Atonement
Why start with Nadav and Avihu? To achieve atonement for the people, it is required for Aharon to go into the Holy of Hollies. Since according to R. Saadia Gaon (verify this) explains that Nadav and Avihu were killed for going into the Holy of Hollies, Hashem used them as an example for what could happen if you go into the Holy of Hollies without the accompanying service.
What sins are being atoned for? The atonement from the service is for all of the sins of Israel.
Is the purification for the people or the place? The purification is for the people exclusively. Any reference to purifying the Kodesh or any part of the Mikdash is not to be understood as purifying the place or objects themselves; instead these objects are facilitating the atonement through the service of the day.
What is the function of various rituals of the service? All the rituals are for the atonement of the people, and even the text says that specific actions, for example:
And he shall make atonement for the holy place, because of the uncleannesses of the children of Israel, and because of their transgressions, even all their sins; and so shall he do for the tent of meeting, that dwelleth with them in the midst of their uncleannesses.
The blood libations, which come to purify the sanctuary, really use the sanctuary and the utensils to purify the sins of Israel
Purification of the Temple
The main focus of the service is to purify the Mikdash from impurity. All issues that deal with atonement are just a byproduct of rectifying the impurity in the Mikdash.
Why start with Nadav and Avihu? Since the death of Nadav and Avihu caused the Midash and certain utensils to be impure, there is a need to purify them through this service.
What sins are being atoned for? It depends on whether the impurity is a Halakhic or Metaphoric one:
Halakhic - There is only atonement for sins directly dealing with the laws of impurity of the Mikdash. (come back) i.e., Entering the Mikdash impure, eating sacrifices in an impure state, etc.
Metaphoric - When Israel sins they are causing a There must be atonement for all the sins of Israel that are causing the Mikdash to be 'impure'.1
Is the purification for the people or the place? The purification is for the Mikdash that has become impure. Any reference to the atonement of Israel's sins is a means to purify the temple.
Is there a greater atonement? The biggest issue with claiming the service on Yom Kippur is solely about purification of the temple is a later verse of the chapter (16:30):
For on this day shall atonement be made for you, to cleanse you; from all your sins shall ye be clean before the Lord.
The verse seems to say that there is a universal atonement that happens on this day which is connected to the service of the day. One way of solving this issue is to say that there is a separate aspect of atonement on Yom HaKippur that has nothing to do with the service done by the Kohen Gadol.2
What is the function of various rituals of the service?
A Hybrid: for the People and the Temple
Both purifications of the Mikdash and Atonement for the people are done in the Yom HaKippurim service
