During the inaugural period of the Mishkan, Aharon was commanded: "Draw close to the altar and do your sin offering and your olah offering" (Vayikra 9:7). Rashi (ad loc.) points out that Moshe had to encourage Aharon to bring the offerings because Aharon was embarrassed. The Ramban (ad loc.) explains that his embarrassment was due to his central involvement in the Sin of the Golden Calf, about which Moshe had asked months before: "What did this nation do to you that you brought upon it such a great sin?" (Shemot 32:21). Thus it would seem that Aharon needed atonement for the sin of the Golden Calf. On the other hand, Chazal tell us, based on the pasuk, "Aharon saw and built an altar" (Shemot 32:5) that Aharon saw Chur slaughtered for opposing the people and was afraid that if they killed Aharon as well, the sin would be so great that they would have no repair (Sanhedrin 7a). His noble calculation, then, seemed to be correct, so what sin did Aharon still have?
The answer seems to be that it was specifically the assumption that the calculations were based on that was the problem. Aharon should have realized that as bad as things could get, Bnei Yisrael could not have reached such a low level that they were hopeless, as the nation has an inner sanctity that does not allow a total moral collapse. In fact, had Aharon opposed their desire to sin from that positive perspective, they could not have stood up to his persuasion.
Chazal learn from a "play on words" on the pasuk "I will place them as heads" (Devarim 1:13) that the fault of the people lies at the hands of its leaders (Sifrei, Devarim 13). This is surprising, as it is not always clear that there is anything that leaders can do to rectify matters. It is understood according to our thesis, for the leaders should believe that the people’s shortcomings are external, not internal. This lack of belief allows the nation to deteriorate further. However, this responsibility of the leaders does not prevent someone like Aharon from bringing his own sacrifice and receiving atonement. The Rabbis learn from "asher nasi yechetah" (Vayikra 4:22) that the generation whose leaders admit mistakes are fortunate (ashrei). One should not concede to spiritual deterioration of any type but should act to reverse it.
From this perspective we can say that, to a certain extent, the stumbling of a tzaddik is a facilitated fall to sin so that he can instruct the people by example how to repent. When people see that he is able to admit sin and return to the correct path, others will learn from him. This is why Moshe told Aharon that he should not be embarrassed. He should admit his mistake and in this way restore the nation’s moral self-confidence and give them the strength to repent.
This is the idea of "Do your sin offering and your olah offering and bring atonement on your behalf and the behalf of the people, and do the offering of the nation and bring atonement for them" (Vayikra 9:7).