In a recent podcast interview, Camille Walsh, author of Racial Taxation Schools, Segregation, and Taxpayer Citizenship, 1869–1973 describes the rise of school funding linked to property taxes in the United States as one of those things that just sort of happened. In effect, it was a structure that appeared to work. So, as schoolhouses across the country merged into districts, the newly formed school boards adopted the same policies. And quickly, it became the norm.
The first places to use property taxes to fund schools were in New England, though for most of the 1600 and 1700s, schools were funded through a variety of means. Some were funded entirely by parents of the children who attended through tuition (also known as rate bills), others by poll taxes, some from general or charity funds, and a few via property taxes. There were even places with combinations - that is property taxes would cover 4-5 weeks and parents would have to choose if they wanted to pay the rate bill to cover more weeks of the terms. In the places where property taxes were the norm, it was generally because there was someone in a position of leadership in the community that had successfully persuaded everyone in the community that it was a net good if the community's children were educated. These property tax laws were typically based on rules and laws in place in England and established during the early Colonial periods.^1
However, there were multiple instances of community leaders explicitly refusing the idea of tax-payer funded schools because simply put wealthier men did not want to pay for the education of the children of poorer men. Which is to say, the idea of educating "other people's children" has long been an issue in American education. The idea, though, didn't come from thin air. Thomas Jefferson was advocating for tax-payer funded public education for the public good in the late 1700s. Others carried on his message and slowly, the tide turned. Rather than seeing paying for other people's children's education as a burden, messaging from common school advocates in the first half of the 1800s were able to persuade communities to establish pools of funds set aside for the purpose of building schools, paying teachers, and keeping the school open.
As you likely noticed from Dr. Walsh's book title, none of this history happened independent of the history around it. During the Great Migration, as recently freed enslaved people and their descendants moved north, previously all-white communities took steps to ensure their schools remained white by setting district boundaries that deliberately cut out neighborhoods where Black families were settling. In some cases, most notably on Long Island in New York State, district boundaries were deliberately drawn to include high-value properties. Meanwhile, in places like New York City, Catholic schools battled in the courts to get access to the city's education funds. Their unsuccessful efforts (plus other concerns they had about the public schools) led to the rise of the parochial system which was funded almost exclusively via tuition and parish funds.
Finally, in a 2010 economic history, Go and Lindert looked at the uneven expansion of public education across the country, especially in the American south. They identified that the nature of property ownership and the structure of school districts meant absent dual systems, white students would be going to school with Black children. In effect, southern states created and paid for 1 and 1/2 systems. They created white school districts and Black districts, but funded the Black schools at one-half or less than the rate of the white schools. Part of this was linked to the notion of who had a right to an education. From their piece: The idea of education rights was particularly linked to taxpaying status in part because education was historically rooted in local tax levies and community decision-making processes and was thus tightly linked to the idea of individual and local “pay-in.”
What this meant is that not only did white lawmakers and district leaders underfund Black districts, they rationalized their decisions based on the amount of money white taxpayers paid as compared to Black taxpayers. (In this answer, I get into the history of rural schools for Black families, including the Rosenwald schools.)
[1] Walker, B. D. (1984). The local property tax for public schools: Some historical perspectives. Journal of Education Finance, 9(3), 265-288.