Earlier in this portion, Aaron's two eldest sons, Nadav and Abihu, attempted to offer "strange fire" before YHVH and, not to put too fine a point on it, were zapped. Moses has told Aaron and his remaining two sons that they are not allowed to mourn, but must "buck up" and get on with their duties. In the fifth aliyah, upon discovering that the sacrifices were not eaten, as they should have been, but burned, Moses once again upbraids them, this time for doing it wrong. Aaron objects on the basis that after what has happened to them, he and his sons are not in a position to expiate the sins of the people, of which we are told, "When Moses heard this, he approved."
It seems to me first that the ritual being carried out improperly is the natural consequence of Moses' refusal to let Aaron and his two younger sons mourn, and perhaps his own anger stems from his knowledge of this. It is also a weakness of concentrating the leadership of an institution in a small, tightly-knit family group. Family tragedies happen, and what is to be done if the entire family is incapacitated with grief? It may be partly a reflection of this realization that when Moses chooses his own successor he goes not only outside of his immediate family, but outside of his tribe.
Secondly, Aaron's point seems to be not so much that he is unable to perform the ritual, but that there is something about him and his remaining sons that makes them incapable of expiating the sins of the people, even if they had done it. Rabbi Judith Abrams talks about the exemption of mourners from certain duties in her first volume of Talmud commentary, and theorizes that the experience of death shatters a person's connection with God to the point that they cannot perform these mitzvot. How much more so must this be true in a case such as Aaron's, where the deaths happened in the course of the performance of these same duties? Aaron is correct, and Moses acknowledges this.