וְנִגְּשׁ֣וּ הַכֹּהֲנִים֮ בְּנֵ֣י לֵוִי֒ כִּ֣י בָ֗ם בָּחַ֞ר יְהוָ֤ה אֱלֹהֶ֙יךָ֙ לְשָׁ֣רְת֔וֹ וּלְבָרֵ֖ךְ בְּשֵׁ֣ם יְהוָ֑ה וְעַל־פִּיהֶ֥ם יִהְיֶ֖ה כָּל־רִ֥יב וְכָל־נָֽגַע׃
And the priests the sons of Levi shall come near—for them the LORD thy God hath chosen to minister unto Him, and to bless in the name of the LORD; and according to their word shall every controversy and every stroke be.
Shenei Luchot HaBerit
I myself feel that the reason this procedure involved the priests (as we read in 21,5) is because the priest represents the attribute חסד. It is this attribute which is required to neutralise the forces of the accuser.