I am not sure if it is just me, but it seems that there is never a melodious chanting of birchas kohanim, but rather is always a mixture of many toneless voices, as if they are tone-deaf. Is this for a reason? Do others share this experience? Should a kohen not sing more melodiously than the others? In Ten Minute Halacha - Looking at Hands of Kohanim During Birchas Kohanim Rabbi Aryeh Lebowitz brings down sources that say that the reason not to look at their hands is because maybe there are blemishes of some sort on their hands and it would be embarrassing for them. Maybe this is a similar reason? That is, to not embarrass kohanim who do not have a good voice? Practical application: Would it be improper for someone to think they know better than everyone else and show off their voice, because really they aren't supposed to sing like that? Does that person deserve rebuke, or are they doing a good thing?
I understand that not all Kohanim (or anyone else) have the most beautiful and melodious voices, but I believe that you are missing the whole point in focusing on this alone. It's the words and deed which matter most, not the voice they come from.
I have seen people pray and sing with the upmost beauty and fervor. Not because they possessed the voice of David HaMelekh, but because they possessed the same passion and devotion as him.
I didn't downvote you, but I understand both yourself and those who did. It's not that your question isn't sincere, it's simply that it's matter is not of great importance.
Peace and blessings brother!
The singing is meant to mark time while the congregants say the prayers in the siddur. (On Shabbos, when those prayers are not said, the singing is omitted.) The Kohanim are trying to mainly stay together and keep a standard pace that will allow the congregation to finish, not put on a show. So if the singing sounds more functional than usual synagogue singing, that is because it is.