Firstly, he's not that confident in his voice.
I don't exactly doubt his ability to carry a tune, but once in a while he manages to sing something in a completely different tune from what I expect and it shocks me because he supposedly can tell notes apart.
During Bounce promotion at Lee Gukjoo's Youngstreet, he was singing Bounce in a cute way and I didn't recognise the tune at all...
But if he actually practises, he could.
During the earlier days, when they went like for Chinese promotions in Taiwan, etc, they actually sang a Chinese song and Minwoo had a part. His voice is mellower but he could carry a tune. Except it was pretty soft in volume, which is most probably because of his lack of confidence.
Minwoo's a bit like that. He only aims for perfection and anything less makes him very unconfident.
That's why he likes to eat up his rap and only perform what he believes he is capable of, and of the rap that he would rap, he'll make sure he doesn't let it sound breathy.
(Though I'd very much hear the entire rap even if he sounds breathy, I'm still holding a grudge for when he chewed on Obsession's rap and didn't let me hear my favourite part in their lives.)