Unfortunately, the question is, in effect, impossible to answer. Due to states' and courts' interpretations of the 10th Amendment ("The powers not delegated to the United States by the Constitution, nor prohibited by it to the States, are reserved to the States respectively, or to the people."), education is a states' matter. Basically, this means there are 50 (plus DC, plus territories, plus the Department of Defense schools, etc.) different rationales for compulsory education laws and public school systems.

There are, though, some general patterns we can look at. First, the original Massachusetts laws cannot be separated from religion. In fact, Protestantism in one form or another helped shape the look of public education well into the 1900's. The New York City Catholic parochial system rose out of a bitter fight between Protestant and Catholic parents, teachers, and school leaders. It's reasonable to say one of the early purposes of public schools was to assist families with ensuring children's spiritual health. However, school would look different in the Quaker enclaves in Pennsylvania than it did in ruralp Massachusetts communities, which looked different than the early academies around Boston that acted as feeders to Harvard.

While schools in all their shapes and forms had multiple goals, they all had at least one purpose in common: Schools were, to put it simply, about teaching children to be Americans.

Nearly to a person, every advocate of common or public schools in America referenced the idea of education for an educated population as a means to ensure a nation of good citizens. The content of this "American" education, though, looked vastly different depending on the children's race, gender, location, and class. One of the first laws related to education on North American soil, passed in 1656 just after the 1647 Massachusetts' law, was a Virginia statue called, "On the Education of Indian Children Held Hostage" which advocated Virginians bring up kidnapped children in "Christianity, civility, and the knowledge of the necessary trades." Meanwhile, other colonies, such as South Carolina, were trying the thread the needle around education of enslaved people. In some cases, slave-owners needed an enslaved person to be able to write or do arithmetic in order to perform their duties. The legislature (with the British government's approval) passed a law in 1711 that said basic literacy, as well as a religious education, was not cause for manumission, even if the enslaved person understood they had to right be free and requested their freedom. Neither of these purposes were about preparing a literate work force or supplementing private tutors or schools.

Which brings us to the public education advocates among the founders including Thomas Jefferson and Benjamin Rush. Both men wrote at length about the explicit connection between education and American citizenship. In their opinion, a voter (by which they meant a white man) needed to understand the issues at hand before voting and the best way to ensure that was to collect and distribute tax dollars for the explicit purpose of educating future generations of men. Unlike the schools across the pond which were deeply segregated by class and religion, they envisioned children from wealth sitting next to the sons of poor or indignant men. While Jefferson focused on the education of white boys (while denying virtually all of the enslaved adults and children he owned an education), Rush was concerned with the notion of "Republican Motherhood" and ensuring white girls got the education necessary to raise future voters. Strong gender-segregation never really caught on America and Rush's arguments helped play a role in assuring families educating daughters wasn't a waste. While Jefferson wasn't able to get his public education proposals passed, his language can be seen in the Northwest Ordinance of 1787.

Art. 3. Religion, morality, and knowledge, being necessary to good government and the happiness of mankind, schools and the means of education shall forever be encouraged.

From that point forward, virtually all new states' and territories' constitutions, in one shape or another, mentioned education and a general commitment to the idea. Back in New England, some states had rudimentary systems of education for almost a century before it was codified in the state's constitution. Others used a state Constitutional Convention to bring in a formal commitment to public education. It's difficult to summarize the Southern states' approach to education in a few sentences but it's safe to say they went to great lengths to ensure tax dollars wouldn't be spend on educating Black children.

New York State and Mississippi provide an interesting contrast regarding purpose and compulsory attendance. First, NYS has the oldest formal system of education in the country, marked by the founding of the Board of Regents to oversee King's College (later Columbia). In 1894, the year the Regents was given control of the states' common, now public, schools, the state enshrined the notion in the state constitution:

The legislature shall provide for the maintenance and support of a system of free common schools, wherein all the children of this state may be educated.

While Mississippi did recognize the idea of common schools in 1934:

The Legislature shall, by general law, provide for the establishment, maintenance and support of free public schools upon such conditions and limitations as the Legislature may prescribe.

A subsequent section, not full repealed until 1977 said, "separate schools be maintained for children of the white and colored races." This isn't to say NYS was necessarily more progressive than Mississippi, but rather to highlight that tax-payer funded education in every state was (and is) tied up with every other political and social system in the state. That is, NYS's vote in 1894 was relatively non-controversial as the state had been doing education for generations. Though there were, for sure, men who were against their tax dollars going to educate another man's child. Mississippi's vote only went through with the conditions related to Black schools.

Their approaches to compulsory education are likewise dramatically different. Numerous states, including NYS, passed mandatory laws about school attendance as early as the mid-1800's. Known as "dead-letter laws" as they weren't enforced in any meaningful way, school was more or less something you did when you could. There wasn't really any social pressure for a child to go to school in NYS until a shift to a per pupil funding structure in the early 1900's. The simultaneous decline in child labor and the rise of a school bureaucracy including truancy officers, meant schools began to care about making sure butts were in seats. It wouldn't be until the 1960's, though, that the idea of being a "drop out", or someone who didn't finish high school, caught on. (There is an entire history related to the Supreme Court cases and the notion of homeschooling and compulsory education.)

Mississippi passed compulsory laws in the 1910's, making it one of the last states to do so. However, the state repealed the laws in the 1950's when it looked like school integration was going to be federally mandated. Those laws weren't put back on the books in the state until the 1970's. Meanwhile, public schools in all 50 states (DC, etc.) could legally refuse to educate a child with a disability until 1975. Finally, it wasn't until 1982 that the Supreme Court ruled all states had to accept immigrant children, regardless of their status, in their public schools. These realities makes it difficult to make generalizations about the goal to educate all children on American soil.

All of this said, the notion that schools are about preparing a literate work force is a relatively modern notion. Known as the "factory-model myth", it's a rhetorical device with limited connection to actual education history. In other words, modern school reformers use the idea that schools are like factories or based on factories or designed to prepare children to work in factories as a rationale for why their idea should be adopted. Like many things in history, there is rarely just one origin story.