Why is the High Jewish Court referred to as "The Cohens and the Levites?"

A question struck me when reading this past week's parsha, a question that irked me all week long. 

"If a case is too baffling for you to decide, be it a controversy over homicide, civil law, or assault—matters of dispute in your courts—you shall promptly repair to the place that Hashem will have chosen, and appear before the levitical priests, or the magistrate in charge at the time, and present your problem. When they have announced to you the verdict in the case, you shall carry out the verdict that is announced to you from that place that Hashem chose, observing scrupulously all their instructions to you."

The term that we use in Judaism for the high court is Sanhedrin (a word taken from the Greek language, according to a friend of mine who is a linguist, Yinon Zaafrany). That said, the high court, elsewhere in the Torah, is referred to as a corpus called "the elders." Overwhelmed by unceasing demands, Moshe, in the desert, was instructed to choose elders. Likewise, the head of the community of elders was referred to in the Torah as the "Nassi," or president, there being an explicit Torah prohibition, "A Nassi, among you, you shall not curse." 

Thus, for me, at least, the complexity, or inscrutability of the highest court being referred to as a body of "Cohens and Levites." All Cohens are Levites, but not the opposite, the same way all squares are rectangles, but all rectangles are not squares. The Torah, though, chooses to accentuate that the supreme judicial body was composed of "levitical priests," perhaps therein hinting at the identity, or nature, of this exalted body. 

Speaking to a sagely friend, a Cohen, yesterday, I gained an interesting insight into the significance of the mention of Cohanim as the epicenter of judicial law. A member of one of the only Cohen families definitively accepted to be undoubtedly Cohanim, Mr. Rappaport, as we lovingly know him, shared that there is precedent for the Cohen as adjudicator; the High Priest wore the breastplate, and served as the intermediary with God to decide matters of utmost importance for the nation as a whole. Likewise, it was only the Cohen who could free someone from the stain of leprosy, purifying him, or declaring his home impure. Lastly, it was the Cohen who prepared the ashes of the red heifer to enable the purification of the nation. That notwithstanding, as we know full well, statistically, the greatest legal minds could not be expected to derive strictly from one small group of priests; the leaders of the Torah world in the last few generations were not Cohanim, and Cohanim, by no means had a monopoly on divine truth, like the people of the Druze or Bahai faiths who believe that divine secrets are to be kept clandestine, and reserved for only the priestly class. 

It would seem to me reasonable to suggest that a Cohen was a person who was distinct, exalted in the sense that his lineage was impeccable, unquestioned. In the tractate of Ketuvot, when a woman was accused of licentious behavior, or having become illicitly impregnated, the Mishna records her kneejerk response, "He (the man) was such and such, and he was a Cohen." It is clear that the man was not necessarily a Cohen, but rather, a Cohen was someone unscrupulously accepted to be a kosher Jew. Furthermore, it would seem that Sefer Hachinuch, in an indirect way, adds greater insight, stating that only a person of impeccable lineage, i.e. "a man whose daughter could marry a Cohen," could serve in the High Court. Fascinatingly, a friend, with whom I spoke over Shabbat, shared that there is a halachic discourse as to whether it's possible to have a Sanhedrin without at least two Cohens, so as to preserve the initial intent of the Torah verses that those who specialize exclusively on spiritual growth are those who attain the most profound knowledge of its nuanced teaches, thus making them best fit to serve in a judicial adjudicatory capacity.