The apprenticeship is for dwarves that have just entered adulthood, as opposed to the schools, which are for children. Apprentices would learn faster, and be less likely to injure themselves, assuming a self-injury mechanic is introduced for unskilled dwarves. As for the idea that dwarves learn fast enough on their own, I don't think it would be too outrageous to suggest that the learning curve for dwarven skills be made steeper. It's kinda ridiculous as it is.
So I guess my proposal is to make dwarves worse at learning, and then to give the player a way to teach them.
I like the self-injury suggestion. I would like to throw in a few others.
1. Mangling the supplies. If a dwarf fails to make something, it should produce "Mangled x" where x is the item they tried to make. Some crafts (such as dying) could simply produce reduced value items while others (such as stonecrafting) could produce waste stone that has to be piled outside the fortress or chasmed. Mangled metal could be melted back down. Once dwarves reach a certain skill level, they stop mangling the supplies.
2. Shoddy goods. Dwarves with low skill may produce a good that has reduced value, labeled as "shoddy x". A shoddy good would be worth very little and probably need to be dumped as well. Producing value should require skill, not just assigning four dwarves to make crappy limestone mugs to buy out caravans.
3. Self injury. To expand on the earlier topic, self injury should range from light cuts to broken bones and even mortal wounds for low skilled dwarves. The chances of this happening would be reduced by skill til eventually dwarves are perfectly safe around whatever sharp object they're using.
This would make apprenticeships useful - getting your child laborers above the danger thresholds for crafting.