While multiple factors play a role - including a whole bunch of social norms around class and whiteness - there are two key norms from education history that contributed to public speaking being a common high school course: recitation and public demonstrations.

First, to recitations. Prior to the arrival of low cost and disposable writing implements and paper, students would typically share their learning with their teacher verbally. (To be sure, during the 17th and 18th century there were plenty of high schools and colleges required lots of writing but even they used recitations as a routine part of assessment.) I'll be borrowing from a few older answers I've written that have touched on the topic.

To set the stage, educators throughout history have struggled with one of the big questions of education in a way not unlike we talk about the falling tree in the forest: if a teacher teaches something and the student hasn't learned it, has teaching occurring? (Educators throughout history have had a wide range of answers to that question.) Which is to say, "I taught it, therefore they learned it" is a flawed construct. There's a whole much longer history related to assessment but basically, if an adult wants to know if a young person has learned a particular piece of information, what's a better way than asking them about it? The concept of recitation covered all of the ways in which a young person would share their learning with a teacher following instruction or independent study. Students would learn new information, by reading it themselves or listening to a teacher/tutor/professor tell them, and repeat it back.

They could be whole class, as in A Visit to Boston Schools, (1856) the author describes:

... in another, the Hancock, for girls, a sister of the Quincy, our visit occurred just at recitation. The teacher gave a slip of paper to a gentleman present, requesting him to write on it the names of several cities or towns in some way noticeable. Meanwhile, he said to the class, "English kings." They at once repeated in excellent concert, "Egbert, Ethelwolf, Ethelbald, Ethelbert," down to Victoria.

Or individual as seen in college entrance exams - a young man was told what to recite... and recite they would for a panel or individual tutor. Such public demonstrations of learning in front of the entire class was the norm at colleges until well into the 1800s. Charles Eliot, who would go on to be the president of Harvard, spoke about his role in persuading the Mathematics department at Harvard to shift from recitations to written exams in 1854. Other departments slowly followed suit.

Meanwhile, a fear of speaking in front of others has been a concern for students throughout history. A 1921 professor wrote about what she described as the "outdated" practice of recitation:

The monopoly of the recitation by the more able and self-reliant students was the next problem to be attacked. This, too, was in part overcome naturally, for many weak and timid students found it easier to express themselves to other students than to recite to the teacher.

All of which is to say, the expectation that a young person would speak in front of their classmates and teachers has been a norm of school going back to the pre-colonial era. Recitation is still used in the modern era - though we describe it differently and have a more robust vocabulary to distinguish what the student or teacher is doing (pop quiz, discussion, memorization) - because it works. While it's true we now understand the specifics better than we did in the 1820s or 1920s - we know about dendrites, neurons, chunking, short and long-term memory - this doesn't mean educators in the past were just doing any old thing without giving thought to what they were doing. They knew back then that sometimes we learn things because we hear, read, or see it and it connects to our background knowledge and becomes part of our long-term memory. Other times, we have to repeat it or develop a mnemonic to get it to "stick."

In terms of the explicit skill of public speaking, it emerged as a stand-alone course for high schools in the late 1800s, early 1900s as the modern high school took form. You can get a good sense of what the course looked like in this guide from 1923. I like this particular book as it reflects the general sentiment about high school between the wars, juuuust before high school became the norm. The authors recognized that high school was still fairly exclusive but the population was becoming increasingly diverse (even if schools were segregated by race.) From the introduction:

Since public speaking is so much used in the higher concerns of men, teachers are more and more coming to see that it is the duty of the schools to foster it wisely, and to guard its quality. They think these ends can be achieved best through the high school . The good public speakers of the next generation will come from the high school students of this generation . They will come almost wholly from the high schools. The high school graduate is a far more favored person than he realizes. Not more than five Americans in a hundred graduate from high school; fewer than twenty-five out of a hundred from grammar school. So we do not exaggerate when we say that the youth who goes through the high school is a very fortunate person, and that he is sure to be a leader in the affairs of the Nation twenty years from now if he is at all active, and worthy, and patriotic. Now and then, some strong, energetic man or woman of great native ability, who has not had a chance to go through high school, will come to the front and be a leading factor in public affairs; but the chances are overwhelmingly on the side of the high school graduate. So, more and more, attention is being given to public speaking courses , discussion clubs , and debating and oratorical contests. Hundreds of contests are being held among the high schools of the various states, and thousands of boys and girls are competing in these contests under the skillful direction of university officers and high school principals and teachers.

The other practice that lead to public speaking as a norm in American schools was the popularity of demonstrations by public school students. Before World War II or so, a community was more likely to be proud of its schools than an individual parent was proud of their graduate. In other words, a community celebrated the school itself, not necessarily the students at the school. This meant schools routinely had pageants, talent shows, bees - spelling and geography, exhibits, and fairs - where students were expected to get up and speak or present.

All of which is to say, talking in front of others has been a part of American public education from the get go. Schools that continue to offer it as a formal course are maintaining that particular tradition.