In my experience, none of my parish priests can sing, but the religious order priests I know can.
My working theory was that men who like to sing join an order for the priesthood, so as to have an opportunity to sing/chant in community, and that those who don’t like this aspect join the secular clergy.
I think the ‘community’ aspect is part of it, but for a slightly different reason.
Before our church was built, my parish attended Mass at the nearby seminary (which was gracious enough to host us). We were also allowed to use one of the rooms for choir practice on Wednesday evenings. On a some rare occasions I would pass by the chapel while the seminarians were chanting Night Prayer. (I don’t know if they always did that or not.) Unlike most local church buildings, the acoustics in that chapel were just right for that kind of singing. It sounded incredible from the outside. (I presume it sounded even better from the inside.) Based on what I heard, those seminarians who were ordained should have been priests with fantastic singing voices.
But it is one thing to sing with a group of people where you have have other voices (and probably a conductor) to help you get the pitch and it is another thing altogether to pull the pitch out of thin air and have to maintain it. People’s voices often crack when they have to speak in front of a group. It’s much worse when singing.
The average choir member I know gets nervous and sings off key when he/she has to sing by him/herself. I even hear that from cantors sometimes. I suspect that even if a priest practices singing/chanting in private, he experiences a bit of stage fright when he’s singing the Mass.
It’s been my personal experience that when I somehow find myself singing off key, I will eventually reach a place where I have memorized the note and realize I am wrong. But I can’t always find my way back to where I should be (at least not before it is time to sing the next note) and then the panic sets in. I seriously doubt I am the only person to whom this happens.
I think most priests, even those with singing experience, need an occasional tone from the instrumentalist to keep them on key. But that requires the priest and the instrumentalist to plan ahead.