Why go to school? To get a career instead of just a job? That's how we might answer that question, but that's not quite the right question for the Greeks. For the Greeks, the less personal question, what purpose does education serve, would make more sense. The contest between public and private would not have been so strong for the Greeks because to be a private citizen was literally to be an idotes. Anybody who was anybody participated in civic discourse and anyone who thought to educate an adolescent would expect them to enter the civic arenas upon graduation. Anyone who taught knew civic discourse was the goal and so taught, however they taught, with that end in mind. (Unless of course they were in it for the money.)
The people who went to school would be expected to be an unpaid or little paid public servants, politicians and lawyers as we would say today. To do so, they would need to know about warfare and trade and history and tradition as well as something of geography and neighboring peoples and customs. But none of these subjects would have anything like the level of specialization they have today and of course the Athenians believed they were superior to all other people, even other Greeks. The goal of education would be to become persuasive in pubilc settings, though toward what end would remain a topic of contention, as did the question of what sort of person a persuasive person would or might be.
Prior to the development of schools, children learned from their parents and then lived as their parents did. Events -- plagues, wars, invasions, famines and other natural disasters -- impacted everyone, but social change was slow. Even when the oligarchs briefly overthrew the Athenian democracy, life went on the same for all but those very unlucky rich whose property was confiscated, like Isocrates. Those people had to find a new way of life, to innovate, we might say.
Isocrates and Alcidamas and Plato were educated by the sophists, Gorgias and Socrates in particular. While the sophists didn't have schools, and made a living kind of like the way TEDTalkers and YouTubers do today, some of their students founded schools, specific places of education which students had to pay to attend. From then on, education became an institution, although not a state intitution until much later, and not just a form of entertainment or source of brief intellectual stimulation. The question from the beginning of these places of education was, what purpose ought schooling serve?
To make young people:
Before you answer a question you need to question the question, right? In this particular case you need to secure definitions if you are debating or trying to discuss in order to come to some kind of shared understanding, which is not the same thing as deliberating with yourself, but perhaps they are allied.
A related question: is there a skill or set of skills the acquisition of which leads to one of the larger objectives listed above? Public speaking, essay writing, creative writing, reading, accounting? Today we would perhaps add coding, statistics, and data-analytics.
Isocrates would say essay writing on politically significant subjects of the day (today a first year composition class with an area of focus like ecology perhaps) for the purpose of becoming an effective citizen, able to arrive at the best course of action both public and private. For him, knowledge is simply a matter of knowing what everybody knows, common sense. The rest, arrangement, style, originality of treatment, are what differentiates the guy at the podium from the guys in the audience. He takes for granted, I think, that learning to write essays on matters of public importance will make you a better speaker while striving to become persuasive will make you a better person because you would know that a person with a good reputation will always be more persuasive than one who has a bad reputation, and so you would always be careful to be seen to do the right things.
For Alcidamas, well, we don't have enough to work with, but if you take "Against Those Who Write Speeches" as emblematic, Alcidamas seems to think that eloquence is the goal of education, but he only offers arguments against writing as his educational platform. How do you become eloquent if reading and writing get in the way? The fact that I've written this down instead of extemporizing it adds weight to Alcidamas' argument that reading and writing ruin the ability to speak effectively in public. Certainly I won't read this to you. But I'm leaving it here for you to read. Hmm? Does he offer any answer to that question? Perhaps one simply is given a topic, thinks up a speech and delivers it on the spot to be critiqued by the teacher and one's peers. This practice existed later, for sure. Often a student would be told not only the topic but also the occasion and to speak (later write) in the voice of a famous person, historical or mythical. Isocrates would approve of this practice on moral grounds because he would say that learning to speak as some great person did would give one great role models to learn from, as well as great thoughts to think -- the great man and later great books tradition. Whether or not Alcidamas would agree I'm not sure we can conjecture. He seems pretty instrumentalist here. Eloquence is the goal, power for whatever purpose the newly powerful person might choose to pursue.
And now that AI can perform eloquent acts for us, in prose and code, what purpose does education serve now?