What a great question! We'll get to the big stuff but there are a few smaller things to address before the main course.

First, it's a bit of misnomer to talk about the "US education system." Due to the 10th Amendment and the absence of education in the Constitution, the schooling of future American voters is a matter left up to the states. In a practical sense, this means there are 50 different systems and history of education in America (technically 56 - 50 states, 5 territories, and Department of Defense schools) with their own debates, influential thinkers, and advocates. Which is to say the concerns in Mississippi in the late 1800's were different than those in Massachusetts. The former was debating compulsory attendance laws and trying to figure out how to fund schools for white children without having to fund schools for Black children. The latter had schools that had already passed their two hundredth anniversary.

Second, regardless of which history we look at, it's difficult to claim the system was "designed." Which doesn't mean people won't try to describe a grand design but people who do that tend to be selling something. Education evolved through fits and starts, over centuries. As an example, some of the earliest documents sent to the place that would become these United States were instructions to colony governors on what to with the Indigenous children. One such letter from 1636 led to the adoption of the Virginia Statues on the Education of the Indian Children Held Hostage in 1656. In effect, European-informed formal education has been around on this soil since before the United States were these United States

These two details (state control, piecemeal development) are significant when it comes to the main part of your question - the concerns of people in the late 1800's. I used the phrase "future American voters" instead of students as it's a helpful way to think about the purpose of American education and what/who informed most concerns. Up until around 1800 or so, education was an ad hoc collection of academies, private tutors, schoolhouses, Dame Schools (basically nursery school), and colonial colleges with the primary focus on the sons of men with access to power - which is to say non-disabled white boys. This isn't to say free Black and African-American children, white girls, Indigenous children, and children with obvious cognitive and physical disabilities never received a formal education but rather, those in power weren't necessarily concerned with them.

Formal education in the colonies and antebellum America was about making future voters smart enough to be informed voters. At the time, the prevailing theory of learning was the that the brain was muscle. So, a young man needed to learn things that smart men knew in order to build up their brain. From a previous question:

[At the time,] the purpose of formal education in these United States was mostly about providing the sons of those with access to power (which is to say white men) the knowledge base common among those with access to power. The curriculum, known as Classical, focused on Greek, Latin, logic, rhetoric, some math, and some sciences. They learned Latin, not because it was useful [for daily usage], but because smart men knew Latin, etc. History, English literature, and modern languages like French and Spanish were seen as less important and usually not a part of formal education.

Meanwhile, their sisters received an education that aligned to the notion of "Republican Motherhood." She learned logic and rhetoric, not because it was important for her future, but for the sons she would inevitably raise up to be good citizens. This, especially in the Northern states, resulted in a system where the children of wealthy men rarely came in contact with the sons of merchants, farmers, and tradesmen. (White and Black children were deliberately and explicitly kept apart. In most Southern states, enslaved people who attended school or were caught with reading materials were beaten or whipped. Free Black children typically attended segregated schools.)

A sea change began in the 1840's. The idea that America would be better off by ensuring all future voters - i.e. white boys - had a shared educational experience began to catch on. In other words, those in power began to see it as a bad thing that poor men and rich men's sons rarely interacted or had a common knowledge base. Wealthy men in towns and cities embraced the idea of school taxes and polls, seeing funded education as a public good.

And now we step back into your question. A driving concern during the 1800s was who was going to pay for public education. The debates were heated and specific to each state. New York State's battle was short and easy - education is mentioned in the state constitution and most villages has a schoolhouse - so education became a public matter without too much fuss. (There would be a brutal and ugly war in NYC schools that would give rise to Catholic schools which I get into here.) In some southern states, the concern over white men's tax dollars going to the education of Black children would continue well into the 1900's.

Taxes weren't the only concern people had. In some places, the marital status of teachers was a huge issue. Some states mandated teachers have a high school diploma, some were unconcerned with her level of education, only her willingness to keep tight control over children. Some parents were deeply concerned with the texts children read in schools, others cared more about how well children presented at the annual school performance fair. That said, there was a common theme that concerned educators across the country in the late 1800s and that was the nature of what children should be learning in school.

During the 1700's and first half of the 1800's, children from families with means would typically learn basic literacy in the nursery or at a Dame school or academy before attending school or working with a tutor for their Classical education. Farmers and merchants, though, weren't especially pleased with the idea of paying taxes for their sons (and daughters) to learn a dead language. Tax-payer funded "common" schools, which were becoming increasingly, well... common and teachers focused on basic literacy and numeracy because it was what children needed to learn. Slowly, via a number of different structures and drivers, the curriculum shifted to English, or Modern. While both the Classical and Modern curriculum are liberal arts, meaning a student experiences a variety of content areas, the Modern curriculum shuffled the deck regarding significance. Greek and Latin were still important, but so was history and literature. Science had been reconceptualized so that children were expected to learn via exploration and discovery, rather than just memorization of words and ideas. The learning theory was shifting away from brain-as-muscle-fill-it-with-any-stuff to brain-as-muscle-fill-it-with-useful-stuff.

Where one stood on the Classical versus Modern curriculum was likely one of the defining issue of the late 1800's. And again, there were other issues - girls were outperforming boys at school and that caused a lot of people concern. Politicans saw a need to "educate" Indigenous children. Black children and white children were coming closer to being in the same space in some areas and Asian children were arriving on the West coast. Hispanic and Mexican children along the Southern border had education needs and the look of their education was a matter of deep concern in some quarters.

In the post I quoted from before, I go into more detail around concern for immigrants and get into how concern flowed in two directions - from white schoolmen and politicans about children and from children's parents. Black parents worked for generations to get access to the same resources white children had. In the early 1900's, there were more than a few Black thinkers who wanted lawmakers to fully fund schools and make real the notion of "separate but equal." Asian parents in California filed lawsuits to get their children access to white schools, which were safer (structurally speaking) and better resourced and staffed than the ones their children were required to attend. Indigenous parents were deeply concerned about the impact of residential school on their children and wrestled with the implications of sending them - or not sending them. Parents of children with disabilities, of all races and ethnicities, were concerned their children were isolated and ignored so they began to organize and collaborate.

All of which is to say every generation of adults has had concerns about the thing known as formal education. And those concerns track right alongside our expanding notion of what we mean by "we the people."