While there are a few eras in education history where I can provide a fairly specific overview of what was happening across the country, alas, the turn of the 20th century is not one of them. In effect, it was one of the most turbulent and idiosyncratic periods in American education history.

The reason for the turbulence comes down to a bunch of factors. New York State, like you saw, had a well-established bureaucracy and data collection system. The state had been doing public education long enough to start a third state-wide reconfiguration of the system (the system was first established in 1784 then reconfigured in 1854, and then again in 1904.) Meanwhile, Mississippi was deep into debates about funding for public schools and that funding hinged on separate schools for Black and White children. This separate system was codified into the state constitution and although the state would mandate compulsory education for all children, the compulsory education laws would be lifted in the 1950s when it looked like school integration was going to become the national law.

Another factor that led to dramatic change was the rise of unions. This played out in a number of ways as it relates to schools, including the eventual elimination of child labor and the rise of collective bargaining for classroom teachers. In the early 1900s, more and more teachers in large cities were filing lawsuits when they were fired for getting married, having children, or otherwise forced out of their job for reasons that were unjust. Women were moving into positions of leadership in professional educational organizations such as the National Education Association and would be speaking at national conferences and shaping advocacy work. Although most administrators were men, the rising profile of women in positions of power began to shift teachers' roles in policy and curriculum decisions.

Meanwhile, curriculum itself was a source of turbulence and wildly different depending on what region and state you're looking at. For the 1700s and the first half of the 1800s, the most common curriculum young people studied in formal education was the classical liberal arts curriculum: Greek, Latin, some maths, some science, logic, and rhetoric. By the late 1800s, the curriculum had mostly transitioned to the modern liberal arts curriculum: reading, writing, mathematics, art, music, history, physical education, etc. The work in the early 1900s was figuring out exactly how much time a young person should spend studiyng each of those things. The 1894 NEA "Committee of Ten" report can give you a good sense of what educators thought was important for students to learn. Note, though, Greek and Latin were still very much encouraged and not a single woman or man of color had a say in the report. The authors were all White men, primarily concerned with the education of their sons and the sons of their peers.

The early 1900s also contained a tipping point around the work to secularize American schools - while some of this work was shaped by the political battles between Catholics and Protestants in New York City, it was also part of a push to Americanize immigrant children. This doesn't mean all traces of Protestanism were removed from American schools - a whole bunch of echoes remain - but rather, there was a collective understanding that the vision of common school meant children of all faiths and classes (But not yet all races and disability status.)

I really do wish I could write you a tl;dr and offer up a pithy summary, but alas, the educational experiences in the early 1900s were so different based on race, gender, disability status, location, class, and religion that it's impossible to summarize.