Is it acceptable for a young child to lead nafl or the daily prayer if they have not reached puberty. If so, give a thorough explanation backed up by reliable references and sunnah.
According to the Shafi'is, adulthood is not a condition for being an Imam. Their evidence is the hadith:
فقدموني بين أيديهم، وأنا ابن ست أو سبع، سنين
Amr bin Salama said: ... They therefore made me their Imam (to lead the prayer) and at that time I was a boy of six or seven years
According to the majority, adulthood is a condition for being an Imam when the congregation has adults and the prayer is obligatory. While it is valid for a child to lead supererogatory prayers like Taraweeh. And it is also valid for a child to lead a congregation of children.
Their evidence is that prayer is not obligatory on a child (Abu Dawud 4403) hence even when he offers obligatory prayer it is counted as supererogatory, and it is not possible to follow him in obligatory prayers. An Imam is liable (Abu Dawud 517), whereas a child is not liable for even himself, then he can also not be liable for others. Also because a child is more likely to make mistakes in silent prayer, rendering them invalid, which is a risk that must be avoided. And it is also based on a report from Ibn Abbas and from some Tabieen. As for the athar of Amr bin Salama that is noted above, it is possible that news of this event did not reach the Prophet ﷺ.
Please note that in the above discussion a child means one who is old enough to be at the age of discernment (مميز).