- Rabbi Avraham Gordimer
One may question how Korach's argument applied to the Priesthood, as Korach seemingly was upset about the general leadership; in fact, his contention (16:3) was against the perceived monopoly on leadership as exercised by Moshe and Aharon, and Moshe in particular was responsible for appointments to all positions of prestige. (To be precise, it was Moshe's appointment of Elitzafan ben Uziel as Prince of Bnei Kehas which stirred Korach's rage [Rashi from Tanchuma on 16:1]. This incident was unrelated to Aharon and the Kehunah.) Why, then, was the Kehunah specially and primarily targeted by Korach?
I think that the answer to this question is found in the haftarah. It is related how Shmuel the Prophet warned the nation about its desire for a king. Shmuel feared that - unlike a prophet, who merely coveys Hashem's direct commands to Man - a king would serve to remove the people from God.
This is exactly the reason why Aharon was the real foe of Korach. Aharon exemplified submissiveness to Hashem. His very "leadership" title was, rather, one of divine service. Korach, on the other hand, sought to usurp Torah and democratize its interpretation, stripping it of holiness and making it into a loose, non-binding text (16:3); he felt that true Jewish leadership meant asserting control on Torah law, thereby disconnecting it from God's defining role. Korach felt that Jewish leaders, somewhat like secular ones, should enact and interpret law as they see fit, in accordance with popular interests, and that avodah was subject to common interpretation rather than Mesorah. Thus, Aharon, whose leadership position embodied pure self-effacement and submissiveness to God, was the true antithesis of Korach's vision for government, and Korach viewed Aharon and the Kehunah in its current state as the greatest enemy of democratic religion.
Although Moshe was also completely subservient to Hashem, his actions were of a legislative nature; Korach objected to the manner in which Moshe exercised his legistlative powers, as Moshe utilized Mesorah, rather than pragmatic and popular needs and desires, as his approach to Torah. Aharon's actions, on the other hand, were pure ritual service as dictated by God; there was no legislative process. This submissiveness to Hashem's dictates and lack of self assertion by one was entrusted with a major leadship position was anathema to Korach and totally antithetical to his vision of Jewish leadership. Thus did Aharon and his position become a prime target for Korach.