Gemara: There was another story, of a non-Jew who passed behind a shul and heard a teacher say: "These are the clothes they will make …" (Shemot 28:4). He asked who has such clothes and was told it is theKohen Gadol. [He decided to convert in order to become Kohen Gadol, came before Shammai with the proposition, and was harshly rejected.] He came before Hillel, and he converted him. Hillel said: Is there such a thing as one who does not know how a king is to act who is appointed king? Go and learn the rules of kingship (i.e., how to be a Kohen Gadol). He went and read. Once he got up to "The ‘stranger’ who draws forward will die" (Bamidbar 3:10), he asked: "For whom is this pasuk written?" He said: "Even for David, King of Israel."

Ein Ayah: [The reason for Shammai’s reaction should be obvious after what we have seen in the past. This piece focuses on Hillel.]
Admittedly, the chance occurrence of awakening to an interest in something good does not prove that a person has a strong affinity for good. However, Divine Providence assures that nothing is by chance, and thus certain occurrences deserve investigation. This is pertinent to the fact this non-Jew passed by a beit knesset when the Kohen Gadol’sclothes were being discussed and that this inspired him to want to convert. This is significant even if he mistakenly thought he could be aKohen Gadol. If he did not have a major inclination toward goodness and truth, the opportunity would not have arisen.
The candidate was not lacking honor as a non-Jew but was interested in an honor that stems from the service of Hashem. This was a sign that the kernel of love of Hashem was in his heart, even if impurities due to lack of education were mixed in. Therefore, Hillel felt obligated to not push him off but to purify his excitement about service of Hashem. His interest was genuine, even though he was used to external flashiness and was unable to settle initially on a simple form of service without attractive aesthetic elements. When he found out that service of Hashem could go together with glory, he jumped at the opportunity to accept service in a manner of adornment. The need for the latter lasted only until he internalized the ways of Torah with its ethical beauty, which in a simple way, contain true grandeur (see Berachot 58a).

The matter of setting aside kehuna for specific people can be explained by saying that it requires exceptional emotion that goes along with the action-based service of Hashem. In other words, it is possible that the actions of sacrifices can be missing the desired elevated emotions that one should have in his heart. One might, then, think that if one reached the level at which his heart is full of elevated emotion and can perform the actions of service with the proper sanctity and grandeur, he can serve as a kohen.
However, it is insufficient to have the type of emotion that a person can reach by choice. Rather, he needs a quality whose origin is determined by Hashem that enables the sacrifices to increase the honor of the Heaven in the desired way and connects man’s actions to the sanctity of Hashem.
The deep connection to the mitzvot applies both to that which one is supposed to do and that which one is to refrain from. That which one refrains from is not only because it is not fit but also because there is a value in holding back from things one otherwise might have done. For example, even if one is on the emotional level of a King David, the pleasant psalmist of Israel, he must not serve as a kohen. Even though, on a certain level, he could do the service effectively, we disallow it due to the gain from refraining from allowing everyone to do so. Only with both positive and refraining from negative can there be requisite purity and sanctity.