There are a couple of different things going on in your question that need to be parsed out. First, the notion of "public" education - that is a system that is funded by tax-payers, free to students, and open to every child* - is a relatively new idea, mid 1800's or so. This isn't the say Rome, China, or Egypt didn't have structures of education. In most cases, pre-modern era systems of formal education were about transmitting what was deemed as important knowledge to the children, mostly sons, of those in positions of power, also mostly men. The prevailing belief was that knowledge didn't have to be applicable to be useful to leaders or those in power. That is, it was believed that learning lots of things made one smart.** If one needed applicable knowledge such as how to run a household or a city-state, that came at a later stage in a child's life or happened via side by side apprenticeships or mentoring. Parents were predominantly responsible for educating their child; society at large, in different ways, would supplement that education.

Regardless of when, where, or who, the general sentiment of formal education for a society's children has been based on the supposition that an educated population was better than an un-educated one. This wasn't necessarily about control, per se. What's often described as ways to control the population can be more accurately described as efforts to manage large groups of small humans, in the same space at the same time. Gathering children together in one place under one adult's supervision required rules and structures in order to be safe. In many cases, those rules and structures were borrowed from the cultural norms of those doing the teachers. As an example, prior to the rise of the common school movement (aka public education) in America, most teachers were men who freely used corporal punishment and public humiliation as a way to control their pupils. When the profession was feminized in the later half of the 19th century, the women teachers used routines from their typically Protestant, working or middle class backgrounds as a way to keep things manageable in the classroom. This meant walking in a single file line, sitting quietly at a desk with hands folded, calling the teacher "Miss" or "Mr", among other things.

So, generally speaking, societies have always collecting together children in order to pass down knowledge. Putting them inside a space designated for the purpose emerged in multiple societies at different times (i.e. the Jixia academy in China, Greek Lyceum, etc.) and took different forms, depending on the needs, philosophes, and structure of said society.

*The most dramatic shift in the modern era around public school is the idea of whose children "deserve" an education. For centuries, education was mostly provided to the non-disabled sons of those in power. The rise of public school expanded the notion of formal education to their non-disabled sisters. Eventually, the idea of "public" expanded to include the children of those who did not look like the men in power, such as Black, Hispanic, and Asian children in America. Finally, the definition broadened even further to include all children, regardless of gender or race, with disabilities.

**There is a great example of this sentiment in Season 1 of The Crown, when young Elizabeth II realizes her education had been entirely impractical. She knew Latin, studied great scholars and thinkers, but realized she didn't have a grasp of modern politics or what she felt was "useful" information. She experienced what is known as a "classical" education and it was the predominate curriculum in American and European education until the late 1800's and early 1900's. Students would learn basic literacy in English and math in grammar or elementary school and then Latin, Greek, maths, and sciences in they continued to high school. Again, the understanding of learning at the time was that knowing lots of stuff smart men knew would make one smart. School curriculum slowly evolved into what's known as the English or modern curriculum, also known as the liberal arts. This meant studying English, history, science, art, music, physical education, and maths - what we basically see in modern schools.