I always appreciate questions like this as they allow for multiple people who study a variety of topics to offer their perspectives. I would echo /u/uncovered-history summary and can offer an additional way of thinking about the differences.

Early public education advocates like Mann were pragmatic and thoughtful about their vision. They recognized it wasn't enough to get people to buy in, they needed people to teach, and ideally, based on his experiences in Prussia, those teachers were well-trained. But also, in a distinctly American move, these teachers would see teaching as practice for their life as a wife and mother or as a replacement for their life as a wife and mother. His contemporaries, such as Catharine Beecher, spoke directly to young unmarried white women and appealed to them to help bring to fruition this idea of a government-funded education for every child. In effect, she positioned unmarried young white women as akin to preachers and pastors, serving their faith in the same way their brothers could from the pulpit. While not all young women who worked as teachers did so for religious reasons, this advocacy helps us understand that Mann and his contemporaries recognized the importance of thinking about education as a system, and not just a personal service provided by one individual to another.

We also see this approach in a shift from the classical liberal arts curriculum to the English - or modern - curriculum that is still seen in American schools today. The classical curriculum was based on the idea that boys became smart men by teaching them the things smart men knew. Their education, especially before the Civil War, came about through an ad hoc collection of tutors, academies, charity schools, and Colonial College feeder schools - basically, by learning from smart men. However, there simply weren't enough smart men to keep up with the demand for teachers a public school system required. There weren't enough donations from local communities to build schoolhouses that were hygienic and safe. They weren't enough parents with home libraries that could be loaned to a schoolteacher. In other words, they argued for public support for a system of public education that would serve every American child, regardless of class.

All of that said, it's necessary to stress that Mann et al weren't interested in every American child. They had limited interest in white children with disabilities. Their vision did not include Indigenous children - they would be "educated" through Residental (our MM post on the schools - English or Spanish) or religious schools. And it surely did not include enslaved children in the American South, who were legally barred from an education. (It's worth noting that Southern states, those who seceded from the United States and joined the Confederacy were among the last states to adopt compulsory education laws and in some cases, the laws would be repealed in the 1950s as desegregation became a possibility.) Meanwhile, common school advocates' interest in free Black children ranged from indifference to open hostility.

And this is where we can see an overlap between the healthcare system and the public education system. Just as it took until the 1980s for public education to become truly "public" (more on that here), advocates for a public healthcare system are pushing to expand Medicaid and Medicare to those who don't meet the criteria set when the system was established. In effect, healthcare is going through the same process public education already went through - the process of defining who counts as a member of the public.

So this is a really interesting story. The big thing to consider is that public education, both in the US and in Europe, developed over a century before public healthcare did in many European countries.

The public education system that we know today in the US is commonly rooted back to Massachusetts in the 1830s. While some cities and locations around the US had begun developing public schools, none had any high standards of uniformity between them. In 1837, Massachusetts saw a man named Horace Mann become the Secretary of Education, for the newly established state board of education, which was in charge of creating a unified school system throughout the state.

Mann was a huge advocate of public education, believing universal basic education was necessary in any democracy. Over his 12 years in office, he would assert that 'Education qualifies each citizen for “civil and social duties”, it teaches “laws of bodily health,' qualifies each person to vote, and provides skills to be a proactive member of great republican country. (These were outlined in Horace Mann’s Tenth Annual Report of 1849). He also helped create a journal for educators, so ideas could be shared between systems. Popularized the idea that 'qualified' people should be educators, which lead to the creation of 'Normal Schools' (such as the Maryland State Normal School,) which existed specifically to create highly qualified and trained teachers.

Soon, nearly every state had boards of education, began creating common curriculums and revising guidelines for who could teach and how. Education became highly politicized during this period, often times with pro education advocates claiming that free public education would be able fix societies problems, especially with the immigrant 'urchin' street children, who many city residents hated. You'll also see that by the 1880s, some European education movements, such as the Kindergarten movement (which was popularized in Germany) spread to cities like St. Louis and then across the US, as new ideas around equation spread quickly. Other additions to public school life, such as cafeterias, were created between the 1880s - 1900, and justification for this was twofold: 1) it claimed that by feeding children, they could then pay attention better in school, 2) we could teach them proper manners that their poor, uneducated parents couldn't teach them.

Ultimately, by 1900, public schools were popular among the general American public, both for US citizens and recent immigrants. It was firmly established as a mainstream part of American society that would continue to grow through the early 20th century when new institutions, such as High Schools, became not only prevalent across the US, but soon became obligatory for students to attend.

On the flip side, public healthcare would not become a mainstream topic in many western countries until the middle of the 20th century, such as when the U.K. adopted it in 1948. However, by this point in American history, many other topics were highly politicized and fought over on the national stage. I don't have as much of an expertise in this subject matter, so I'll hold back a more detailed answer for another expert to weigh in. However, the point I want to emphasize is that by the time healthcare as a universal issue became a topic, US public schools had been established for over a century, so it wasn't something on the table that people were focused on or would ever consider getting rid of.