The Vocal Fach System was created to describe the kind of voices that were needed to sing specific roles in OPERA. Well-trained opera singers can often cross fachs--singing lyric OR dramatic roles for example. Other singers may specialize in one type of role or another.
It is both pretentious and inaccurate to ascribed terminologies like "lyric baritone" or "leggiero tenor" to non opera singers since the terms apply to UNAMPLIFIED natural voices. A "light" voice in opera would NOT be considered a "light" voice in pop, because that voice would have to have the penetrating power to be heard over an entire orchestra in a huge hall without the benefit of a microphone.
I only heard one song by Hozier, but just call him a baritone and leave it at that.
Vocal ranges can vary in different singers of the same voice type, and two singers can have very similar ranges, but belong in different categories due to other factors.
According to Wikipedia, Hozier does have some formal training as a musician, but it doesn't indicate if this training was vocal or not (at any rate, he does sing in a pop-style, not classical), but it mentions he was a member of an orchestra.
http://en.wikipedia.org/wiki/Hozier_%28musician%29
Before, he was a member of Anúna, an Irish choral group which in addition to SOME traditional music, sang original music meant for a combination of trained and untrained voices.
http://en.wikipedia.org/wiki/An%C3%BAna
I found Hozier singing solo for "Le Chanson de Mardi Gras" (in French). This is not the way a trained singer would sing, since there is some sound of straining on those top notes that would not be there in a trained singer. He still sounds like a baritone to me, but he is also very young here. Also proper training can sometimes "open up" parts of a singer's vocal range that they may not use because they literally don't know HOW to use it.
https://www.youtube.com/watch?v=RqabVDNrhlY#t=70
Basically, I actually think "Take Me to Church" is catchy. I don't regard it in the same sense as opera (and I'm a big fan of opera), so neither should you. Sometimes it's just best to take music at face value and not to try to shove it into tiny boxes. Would it make Hozier a better or worse singer if you labelled him?