What books were they teaching in high schools back then rather than 1984 or Of Mice and Men?
The easiest way to think about the shift of literature from the 19th century to the 21st is to consider the shift in the student population. For most of the 1800s, school was a thing white children from working class, rural, or poor families did if they could - didn't do if they couldn't. School was held for a few weeks at a time, in the Winter and Summer and it wasn't an especially fun place to be. Students typically left when they were done, not necessarily when they graduated. Meanwhile, Black and African children in the American South were barred by law and practice from learning to read, much less attend school. Black children in North typically had to walk past several white schools to attend an African Free school and Indigenous children were routinely taken from their families in order to be "re-educated" or "civilized."
If we focus in on high school, though, we're talking about the education for America's future leaders, politicians, bankers, religious leaders, and lawyers - which is to say white boys. In the 19th century, they would experience what's known as the "Classical" curriculum - Greek, Latin, some math, logic, rhetoric, and science. What was missing from their curriculum, though, was literature. At the time, the study of literature and history were both seen as cognitively less demanding that the other subjects. The prevailing belief was that a young man got smarter by learning hard things. Reading literature (or studying French or other modern languages) wasn't hard in the way Greek and Latin were. So, basically, no one was teaching literature in high school until almost the end of the 1800s.
By the end of the century, high school was increasingly something children of all genders and races did. High school attendance rates were still low as compared to today but most major cities had several public schools and one high school serving a large rural area wasn't uncommon. As the popularity of high school rose, so did the curriculum. Teachers and parents advocated for a more modern curriculum which would provide more practical content for young people. By 1892 and the National Education Association of the United States Committee of Ten on Secondary School Studies, teaching literature as a part of the school day was no longer seen as something frivolous. The English committee wrote:
The intelligent reader of the report of this Conference will find described in it the means by which the study of English in secondary schools is to be made the equal of any other study in disciplinary or developing power.... The main direct objects of the teaching of English in schools seem to be two: (1) to enable the pupil to understand the expressed thoughts of others and to give expression to thoughts of his own ; and (2) to cultivate a taste for reading, to give the pupil some acquaintance with good literature, and to furnish him with the means of extending that acquaintance.
So, the literature that was read in American high school English classes until 1950 or so was very much part and parcel of the canon. Some teachers, though, would advocate for an even more modern reading list and suggested books like:
I swear I did not pick and chose that list of books. Rather, I copied and pasted it directly from Mary Esther Pfeil's 1908 Master's Thesis, "The English Curriculum in Secondary Schools", page 44. I suspect, though, you noticed the pattern about the focus on the books. Students overwhelmingly read literature authored by white men about the lived experiences of white boys and men.
The texts began to change in the 1950s and 60s, as the student population changed. First, schools were desegregated following Brown v. Board. (Similar to white schools, Black schools could go with a classical or modern curriculum. Several cities had Black high schools with a classical focus, most notably in Washington DC where Black schools receive the same amount of funds as white schools.) This desegregation meant children of all races were learning together, and given the increasing focus on the "modern" or practical curriculum, teachers began to expand the nature of what students read. There was still a focus on the canon but teachers began to incorporate more popular fiction.
Which leads us to the massive sea change in the 1950s brought about through social change, college admission practices, a shift in who became teachers, and the increased accessibility of affordable class sets of paperbacks. From a previous answer on the question about political books of the era:
... high schools that wanted to brag about where their students went to school would focus on texts they knew appealed to a particular college admissions' officer. This feedback loop lead to the creation of lists and programs, including Great Books and Junior Great Books as well as the National Conference, a group of university and college representatives who established the list of classics that students would need to know and be able to write about for their entrance exams.
Concurrent to these changes around curriculum, the teaching force itself was changing. For generations, it had been a temporary thing a young woman (mostly white) would do before getting married and having children. Much like the push for voting rights, women challenged the notion a woman couldn't teach and be a mother and that the job of being a teacher required professionalism. However. America being a country with deeply sexist roots, well-meaning schoolmen felt teachers needed guidance and help. This sentiment lead to the creation of professional education organizations, such as the The National Council of Teachers of English. Founded by James Fleming Hosic, it was established as a roundtable working group of English teachers and members of the National Education Association in 1911. Unlike the 1894 Committee of Ten, the working group included practicing teachers, many of whom were women (who wouldn't be trusted with leadership positions for years.)
Increasingly school leaders pushed back against the idea of colleges having say in what students in secondary school did and one of Fleming Hosic's goals was to advocate for high school curriculum written by high school teachers based on their goals for their high schoolers - not college admission criteria. Slowly, high school became less a place to prepare for college and more a place that teenagers went. NCTE held (holds) annual conferences where teachers would model curriculum, share lesson plans, reading lists, and diffuse knowledge about English class across the country. As some districts linked teachers' evaluation to participation at such conferences, attendance was high and lists were widely circulated.
There's one last shift that needs to be addressed before we can get to the particular books on your list and that's the idea that school should be a place that is accessible and welcoming to young people. Progressive education advocates suggested the radical notice adults should consider children's interests when planning curriculum. This notion, spread through conferences like NCTE, meant teachers were increasingly choosing whole class books that reflected current events or ones their students wanted to read. This lead to a shift in what's thought of as canon - all of the texts you listed are now considered part of that canon and have appeared on an NEA or NCTE list or conference session.
So, one last feedback loop. Young people read those books in school, grew up to become teachers or curriculum directors, and teach what they were taught.