Support, but with reservations. More important (IMHO) than allowing child labor, or even allowing children to simply observe labor, is to evaluate the ages at which work can be done, on a job-by-job basis. I don't know which is more ridiculous, slapping full plate armor onto a 12-year-old and telling her to go fight goblins, or pretending that an 11-year-old can't milk a goat.
Granted, these are dwarves, not humans. Dwarven lifespans are almost exactly twice a human's, but then dwarven babies learn to walk a bit sooner than humans, so their age progression is not simply "Human x2". For now, I'd like it if all of the professions were given a Minimum Age rating, which took into account their difficulty, requirements in craftsmanship & attention, chance of injury, and of course requirements in size & strength. I can easily see children as young as 5 years old doing really poorly at things like Growing, Weaving, Milking, Cooking, Pottery, Wax Working, etc. But they should be around 20 years old before they'd be capable of doing the real heavy work like Mining, Blacksmithing, Masonry, Woodcutting, and of course the military, or the more exacting work like Gem Cutting and Suturing. The child's individual size and strength could/should also be taken into account, and affect "eligibility times" accordingly.
But this thread is about assistants, not workers. I really like the idea of using them to fetch things, although instead of food & drink (they can't carry drinks anyway, unless they're strong enough to take the whole barrel), I would have them collect the job components. So the instead of having the Legendary Furnace Operator run around collecting the iron, pig iron, charcoal, dolomite, and charcoal (which can take all day if you have far-flung stockpiles), he can simply say, "Gophers! Go fer." While they're away, he can either talk to his remaining students (if any), or just stay in his workshop On Break, deducting (at least some of) the time thus spent from his next scheduled break. Professionals with fewer apprentices would help collect the items themselves, like an Armorsmith picking out her next steel bar for herself, but sending her one helper for the much less important bar of coke.
Children should select their masters based on their preferred working materials, if working for that master would actually bring a chance to work with that material. For example, a child who likes giant jaguar tooth might gaze longingly at the Legendary Bone Carver's handiwork, before heading to the Carpentry shop to try her hand at working with willow wood instead. They should also have a predilection to their parents' professions, if their mom & dad does something they can learn at a young age. If not, they might apprentice in a related field first--Engraving, then Stonecrafting, then finally Masonry or Mining. But it's the master who actually decides to accept the student or not, and of course the overseer can boot apprentices (as well as deciding how many pupils a master can accept). But I think it should be the apprentice who has to initiate contact, and not even the overseer should be able to force that. After all, a child does as it pleases.