One of the things we know for sure is that every place that has created a tax-payer funded, compulsory system of education has followed its own timeline and had its own purpose. That purpose differed between countries, states, and towns and in many cases, it's more accurate to say there were multiple purposes.
A common sentiment is that tax-payer funded, compulsory schools were created to train a docile and compliance workforce (or army - more on Prussia here). I get into the problem with that narrative in a response to an older question but what's compelling is that the "factory model" origin story is often accompanied by the idea "summer break is because of the agrarian calendar." And yet, neither is supported by the historical record and fall apart under scrutiny. For example, there are instances of schools inside, or next to factories, in large urban areas around the turn of 19th century. Students who attended the school often worked in the factory itself. Which raises the question as to why a system needed to be created to train children to do jobs they already had. In addition, tax payers were funding public education in places without any large factories to speak of, such as North Dakota and Colorado. Meanwhile, nothing usual happened on a farm in the summer that would require a nation's children; most of the work is done in the spring and fall. And it wouldn't make sense to create a system that trained children for factory work but was organized around farming calendars. More to the point, before the American and Canadian school calendar evened out in the modern era, schools often held summer (and Saturday) sessions.
If forced, though, we can bring all systems back to one purpose: large societies have found that a designated place to bring children together under the guidance of a trained adult is an effective and efficient way to transmit knowledge from one generation to the next. In some cases, that knowledge originally had a spiritual or religious focus (i.e. Massachusetts, Quebec, Finland) but has evolved into a secular curriculum. In other cases, it was secular from the beginning (i.e. the Netherlands.) Some countries, such as Saudi Arabia, have maintained a religious focus. The modern British government school system is built on and around charity and private schools, which often, but not always, had a religious focus. Several Asian countries can trace their education systems back to the 5th and 6th centuries, including government funded schools for village children.
The notion of "compulsory" education has likewise shifted over time. In the modern era, "compulsory" education is generally more about a government's responsibility to children than a child's obligation to attend a government-funded school. The United Nations Convention on the Rights of the Child, signed in 1989, says:
Article 28
- States Parties recognize the right of the child to education, and with a view to achieving this right progressively and on the basis of equal opportunity, they shall, in particular:
(a) Make primary education compulsory and available free to all;
In effect, this means that governments cannot pay for education for boys and not their sisters, for non-disabled children and not for children with disabilities. While most countries have had some form of education for children with disabilities, it would have been inconceivable for those government officials who argued for funding public education that children with physical, cognitive, or emotional disabilities would be in the same schools as their own children. Basic literacy was illegal for most enslaved people in the American South until the end of the Civil War, which meant schools were off limits to them. So while there were compulsory education laws on the books in Massachusetts and NY, Black children in Alabama could be beaten if they were discovered to be literate. In many cases, once school was an option for Black or African-American children, white taxpayers would refuse to pay school taxes without assurances their money wasn't going to fund Black schools. Likewise, some schools in the Southwest were off-limits to children of Mexican origin while an entire system of schools across the country were attended only by Indigenous or Native children, often taken from their family by those who ran the schools. To make it clear: the purpose of school for those children was vastly different than the purpose of school for white children in Boston.
There is one last concept to discuss and that's the idea of the "grammar of schooling." Tyack and Cuban, two educational historians wrote about how, despite a variety of starting points, reasons, funding sources, and populations, school around the world in the modern area has developed a sort of language. This grammar includes things like desks in rows, colorful wall decorations, separate bathrooms for boys and girls, and calling teachers by a gender identifier (Ms., Mrs., Mr.) and their last name. This grammar has spread, not so much because of a school's origins, but because the idea of "schooling" in the modern era is represented by certain touchstones. Not all schools embody all of the grammar and sometimes, the things that make school recognizable as a place called "school" stick around not because they're helpful for teaching and learning but because they're familiar. So, it's not uncommon for us to look back from 2019, see examples of the "grammar of school" and reasonably conclude they all come from the same place.