I have to defer to those who are more familiar with the histories of education outside the United States as to why they developed uniforms but the simplest history for why America didn't is, basically, no one cared enough about uniforms to advocate for them.

Pulling back a bit, there are two key things to highlight. First, there is no national education system in America. What this means in practice is that nearly everything that happens regarding public schools happens at the state or local level. Second, despite this lack of a national system, there is a pool of things - behaviors, objects, words, etc. - that identify American schools as being schools. This pool is described by education historians as the "grammar of schooling" and includes things like apple motifs, calling teachers by a gendered title and their last name, boys and girls being educated together, etc. - but does not include uniforms. There is the concept of a school "dress code" but the specifics of codes vary from school to school and are often linked to adults' perceptions about what's "appropriate" for children in the community to wear. While dress codes could be seen in school policy documents in the early 1900s, it wasn't until after World War II and the arrival of the Baby Boom generation in high school that they became part of the grammar of schooling. There was a push for uniforms in the mid-1990s but it didn't catch on in any meaningful way and there's no reason to think they ever will.

It sounds fairly flippant to say that it's most likely the reason American public school children do not routinely wear uniforms is that no one cared enough to advocate for them but that's pretty much the whole ball game. In contrast, early common school advocates cared deeply about the gender of the person in front of the classroom and we see the consequence of that in the feminization of the profession in the mid-1800s and continued demographic patterns. Likewise, communities and advocates cared deeply about the role of public school buildings as evidence of a community's commitment to education - which is why many communities built large, sprawling schools on the outskirts of the town or village. We also know public school advocates saw the value in a modern liberal arts curriculum (math, science, history, literature, arts, physical education) over a classical liberal arts one (greek, Latin, some math, rhetoric, etc.) and by the 1940s or so, the modern liberal arts curriculum had mostly replaced the classical one.

It's hard to know exactly why advocates didn't push for student uniforms but it's likely that class played a role. One of the most comment arguments for public schools is their presence meant the sons (and daughters) of wealthy men would be sitting in class next to the sons (and daughters) of poor men (and their wives.) While they carried over a number of touchstones from England, early formal education advocates in the colonies and later the United States likely saw school uniforms as something that indicated the upper class. Brunsma's 2006 book, "The School Uniform Movement and What it Tells Us about American Education" is the most comprehensive look at uniforms in American schools and he starts the history at British schools such as Eaton and describes the uniforms, badges, and other accouterment British schoolchildren wore, as an "unseen, insidious mode of social control." It's possible the American men who pushed for a public school system in America wanted to avoid such social coding.

Another reason they likely didn't push for uniforms was the informality of attending school. The idea that nearly every American child would spend the hours between 8 AM and 3 PM or so, Monday through Friday, in a school classroom didn't become the norm until the 1940s. It didn't become the norm that they would stay in school until they were 17 or 18 until the 1960s. Before that, children went to school when they went to school. And if they didn't... they didn't. This meant, practically speaking, that children could go from chores on the family farm to school and back to chores. Donning a particular set of clothing just to attend school wouldn't make a great deal of sense to children - or their adults - as there was no real social pressure attached to attending, or not attending, school.

All of that said, there were formal educational settings where children were expected to wear particular clothing AKA a uniform. The parochial school system - which is to say private, religious schools - rose out of series of conflicts between Protestant and Catholic parents, school leaders, and community members. As Catholic parishes established schools devoted to a Catholic education, they developed their own grammar that served to separate their schools from the increasingly secular public schools. Included in their grammar were expectations around uniforms. Over time, much like it did in England, different colors or badges came to represent different private schools. Many secular private schools adopted similar policies, their uniform choices acting a signal of the young person's participation in a particular community. Meanwhile, young men who attended private military schools were likewise expected to wear particular clothing while they were at school. As Reserve Officers' Training Corps (ROTC) and Junior ROTC programs became more common in public high schools, participating students were (are) expected to wear their uniform to school for some or most days.

Another school setting where children would be expected to wear a particular set of clothing were at the Indian Boarding Schools. You can read more about their history [here] (https://old.reddit.com/r/AskHistorians/comments/np9lez/who_is_this_child_an_indigenous_history_of_the/) and in contrast to the religious or secular uniforms, the clothes (and hairstyles) children were expected to wear were about breaking the child's bonds with their home and communities, not with strengthening them. While the uniforms worn by Catholic schoolchildren indicated they were part of a particular community and culture, the uniform Indigenous children at the schools were required to wear was about signaling they had moved away from the clothes and hairstyle of their families and were adopting the look of the domoniate white culture.