The Tosefta records how the Kohanim would blow trumpets to rally the people and gather them for Hakhel.
When Moshe teaches the Mitzvah of Hakhel, he was addressing the Kohanim and Zekeinim. Based on this, some commentators write that the reading of Hakhel was performed by one of the Kohanim, possibly the Kohen Gadol. The historian Josepahus records that the reading of Hakhel was performed by the Kohanim Gedolim.
The Mishna teaches that Aharon, the first Kohen Gadol, epitomised the quality of Ahavas Yisroel. According to Chassidus, this quality of Chessed and love is the defining characteristic of all Kohanim.
The Mitzvah of Hakhel was about gathering the Jewish people together, which reflects the theme of Ahavas Yisroel.
Further, Hakhel was about inspiring the people in the observance of Torah and fear of Hashem. The approach of bringing Jews closer to Hashem and to the observance of Torah, must be approached with love - אוהב את הבריות ומקרבן לתורה as embodied by the Kohanim and "the disciples of Aharon".