Can you say a little more about your question? That is, I can shed light on the progression of compulsory education laws and the rationale behind them but I'm not sure if that's what you're looking for.

I am not sure I can understand your entire question, but I can answer some of it. Apprenticeships stopped because of an economic change, and also because of a legal change. The economic change was the Industrial Revolution, when more and more small shops were displaced by factories, big shops. Factories used skilled and semi-skilled workers, but those workers would be trained within the factory because that's where the equipment was. It was pointless for a farm girl from Vermont to serve an apprenticeship with a weaver in order to learn to run a row of powered looms in a textile mill in Lowell, MA. It should be said that an apprenticeship was just part of a much larger traditional practice of sending children off to live with other families when they were in early adolescence. Girls would be sent to live in another household, sometimes households would exchange children. This was to make them learn how to be obedient ; to learn to take orders and to serve. With industrialization, that system was disrupted.

The second change was legal. An apprentice was an indentured person. In return for learning to be a blacksmith, say, he would be forced to serve as an employee for some years. The first couple of years he would be learning , and as he acquired skills he would be more and more cost effective until the last couple of years he would be a skilled employee., working for free. The master blacksmith and the apprentice ( typically around 14 yrs old) would have entered into a contract for this; the blacksmith was to have taught him the trade, fed and housed him, made sure he went to church, seen to his basic education in writing, arithmetic. The apprentice was supposed to be obedient and sober, apply himself to his task and learning, and not get married. But there was a strong incentive for the apprentice to run off, once he'd gained enough knowledge to be skilled. And there was a strong incentive for the blacksmith to keep the apprentice somewhat ignorant, badly fed and hard-worked, so he could have an inexpensive employee for several years who would not go out and compete with him. And, at the end, the apprentice was not free. If he ran away, before his contract was done, he could be apprehended and returned- like any indentured servant. That was pretty much stopped in the US in 1833, when imprisonment for debt was done away with ( and so a runaway apprentice could not be jailed for his debt to his master), and decisively after the Civil War, when indentured servitude was eliminated as being , essentially, slavery.