One feature of American schools that's a little hard to wrap our heads around in the modern era is that a school was built for a community - not necessarily the young people in the community. It feels a bit inside-out but it's that's more because of how we think of schooling, especially high school, today.

The first piece of context that is helpful to understand is that the earliest schools were very much in service to the individual child. In fact, the earliest schools for white children in New England were based on straight numbers: if you have more than 10 children, you need a schoolmaster. More than 20, you need to a schoolhouse. (Or variations thereof.) These laws stemmed from the idea that white children were entitled to basic literacy skills as preventative measure against evil forces and the community shared that responsibility with parents.

As the new country took shape, though, the sentiment shifted a bit. Founders, most notably Jefferson, advocated hard for a community's responsibility to educating all of the children in the community in service to an educated population. In effect, the thinking shifting from "we need schools to teach these specific children" to "schools are a good idea for a community to have because a community has children who grow up."

By the mid-1800s and following a hard fought PR battle by the likes of Horace Mann and Catherine Beecher, "common" schools were the norm across the country. While we think of these schools as "public" in the modern era, the concept of "common" was meant to evoke a shared sense of obligation to education and the country. By the end of the 1900s, states had thousands - if not hundreds of thousands in the case of NYS - of schoolhouses. To borrow from an older answer on school construction:

... school was increasingly something children did in service to a public good. This meant that towns and communities increasingly took pride in their schools. High School graduations, even for a handful of students, were marked by large celebrations by everyone in the town. Thousands of people would turn out for recitation events where children would rattle off passages from American texts they memorized. This meant as towns and villages built new schools, they were more likely to select locations that were aesthetically pleasing - a point of pride, as it were. Rather than building a new school in the middle of town, two schools might be built on either end of town, perhaps overlooking a bluff or on a particularly attractive patch of land. Given there was more space, they were more likely to build out, rather than up.

Then, came the depression.

The second wave [of school building construction] crested with the creation of the Works Projects Administration (WPA) and the New Deal. School construction had slowed up considerably due to the depression (as an aside, this is when many districts passed laws banning the employment of married women. Unmarried women could teach but once they got married, they had to leave as district leaders wanted to protect jobs for married men.) By the 1930s, school architecture had become its own branch of the trade and architects designed large, open spaces with lots of natural light, multiple entries, curving staircases, and Grecian or Roman columns. (To be clear, though, this type of design was for white schools. There were some Black schools that were built during the WPA but in the cases where the school was more than a functional building, community members typically had a role in funding the aesthetic changes.) The WPA funded the construction of thousands of schools across the country, added auditoriums, murals, and gymnasiums to others and helped establish the look of the modern sprawling suburban American school. The WPA is why there are some very rural places in the country with absolutely beautiful testaments to public education.

And what community wouldn't be proud of such a building (and the children in it - including who was included and who was excluded)? I'm not familiar with World War I's direct impact on school spirit but I can say that same time frame corresponded with a period of consolidation; school houses merged to become school districts. And school districts needed names. (More on school and district naming practices if you're interested.) Schools had always been places for public performances and musical and theatrical performances continued that tradition - communities would turn out to see students perform. They would also turn out to see them play sports.

To borrow again from another answer - this one about mascots based on Indigenous imagery - school administrators, mostly white men, were a professional class, which meant a great deal of networking. They frequently belonged to schoolmen clubs where they would share ideas and resources and the rise of athletic leagues at the college level shaped how they thought about their own sports teams.

So, to recap: communities built schools they were proud of. This pride became part of school climate and culture, become more focused and specific with the consolidations of schools into districts and the creation of sports leagues. There was, though, one more "event" that shaped school pride: The Civil Rights movement. The 1950s saw not only the arrival of the children born during the baby boom at schools, it also saw the forced desegregation of schools. In many communities, white adults felt it was very important to clearly communicate who a school was for and who was allowed to have pride in the school and the school name. That sentiment is how several states ended up with high schools built in the 1950s named after Confederate generals.