Is there a reason for not training Nezuko in breathing?
Nezuko sleeps not because she has nothing else to do but because in such a way she restores her strength instead of eating flesh. She slept through Tanjiro's training year, while (probably, not shown really) he tried to wake her up.

From the slayers' point of view, they most likely have no idea how to train demons because they have a different physiology. (Upper Moon One seem to have mastered the moon breath before becoming a demon.) Also, pillars still may think it is not safe.