I wrote a pretty decent, but still rambling reply yesterday then posted it as Reddit broke.

I'll try to be more succinct this time as I've had more time to think about it.

I like the idea of there being a distinct difference between schooling and education. For mysel,f if I had to define the terms, I'd say that education is the acquisition and experience of knowledge and skill, whereas schooling is education minus experience. As a metaphor, schooling is a photo from the summit of a mountain, where education is climbing the mountain and standing on the summit where the photo was taken.

Your definition of what an education should be is far more holistic than I think I would personally define it, but I think it encompasses the fullness of what society thinks of education. An education has always served multiple purposes:

• To obtain mastery of a skill. This is, I think, the primary reason people participate in education later in life. An education isn’t a single process, but a continuous one. In older times, people would engage in an education of the basic skills they needed in their day to day lives, and as society became more advanced, specialization occurred due to the rise of industries, which required people to become less educated in a number of skills and focus on one. While a general education in a number of useful skills would still be beneficial, the social ecosystem had changed such that generalists fared more poorly than those with a specific set of skills who could prove more useful to the society in which they lived.
• To provide a basic level of knowledge for functioning in society. This would be things like literacy and numeracy. Industrial society doesn't function well if most people can't read or add, and doing so makes a person more valuable and productive to the systems that run our society. Industrial society gave way to our modern society where knowledge-based professions (i.e. white collar) became more numerous, diverse, respectable, and better paying. Mastery of a niche knowledge profession, while it takes a literal lifetime to do, is still the most certain route to building wealth for someone who started with little. This situation has contributed to something of a knowledge arms race in the population, which leads to the next point.
• A better future for one’s children. Before formal education existed, most people didn’t have an opportunity to receive an education. As children, they were expected to join the family in whatever chores or business that their parents had to attend to. Others would be pressed into industry where their education would consist of operating specific machinery or factory work, others might be sent off to learn a particular trade as an apprentice. Very few people until the modern era were able to gain a formal education and those that did so paid for private tutors or sent their children away to boarding schools. As wealth began to spread among the population with industrial development and more individuals could join the ranks of the middle class, an education became a means for future generations to escape the hardships foisted on older generations and a greater chance at surviving to adulthood. There’s a reason that child labor laws in the US began to be passed at the state level at the same time elementary school education became compulsory. 
• As a means of improving one's class, status, or social standing. As the upper echelons of society gained more free time, education became a way to set themselves apart from commoners and they could use that free time to pursue an education. As the Enlightenment wore on, the benefits of an education became more pronounced and more desired. Scientific discovery and invention became a source of status in itself. Common people who desired to join the ranks of the middle class or have their children do so considered an education as the means to achieving that. The other source of status that became more accessible was that of the arts. Becoming an artist, whether that be a painter, musician, or writer was, by the time the Enlightenment period was in full swing, a far easier prospect than it had been before and people could achieve a level of fame and therefore status that previously was unknown to people of an earlier time period. Likewise, being able to participate to some small degree, whether that was playing an instrument, being well read, or being able to enjoy an Italian opera, was seen as an indicator of being well raised and therefore a better person. The common people, seeking the trappings of status and the upper class, desired to learn the arts to be able to participate and enjoy those cultural artifacts and even engage in them with the hopes of increasing their own social standing.
• To imbue a new generation with a set of cultural values. This is seen best in religious schools, which are common in many places of the world and throughout history. While a religious school teaches little that is useful about the world, it does inculcate a new generation with a sense of identity and cultural values. The society at large finds it useful to continue the practice as it creates social cohesion and a distinct identity from which to find common cause and common bonds. Under supervision from instructors, the new generation is socialized to participate and engage those values in a setting that reinforces them such that when they leave, they can reinforce those values on the community at large.

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