I've had this question open in a tab since it first appeared, trying to figure out the best way to answer such that I attend to AH rules around depth without giving you 30,000 characters on the history. So, I'm going to offer context, which will likely differ from someone who is more familiar with the economic history of school funding.

The first thing to establish is there is no national education structure in the United States. What this means in a practical sense is that the history of vouchers in one state looks very different than it does in a different one. The second marker to put down is that the history of both of these topics started decades and even centuries earlier and the starting point a person picks is, generally speaking, reflective of their political and/or philosophical leanings. These two details mean there is no "clean" history of vouchers and charter schools.

One history would tell you that by the late 20th century, the idea of education as a personal good, as opposed to a public good, was firmly anchored among the public. Libertarian-minded authors such as John Taylor Gatto meandered through education history, picking and choosing details for his books in order to add to a chorus that public education was fundamentally bad for the individual, and as such, a dramatic intervention was needed. Other members of the chorus included politicians (mostly Republicans but some Democrats) who released reports comparing international test scores and making all sorts of proclamations, focusing on the idea that schools were "failing." A not small number of American voters accepted this idea that particular test scores meant schools were failing and were amenable to solutions that included adding competition to the mix. Conservatives would point to the array of choices available to parents before the rise of common schools (academies, private tutors, private schools, church run schools, etc.) as a good thing and argued that having a diverse system of school options for parents - supported with public dollars - was better than a single system that requires parents to pay twice (taxes and tuition) if they picked something other than public school. Advocates of this particular history would likely say that the idea for vouchers and charter schools started with Milton Friedman in the 1950s and what we saw in the latter half of the century was the idea reaching critical mass.

There is, though, a different way of looking at the rise of "alternatives" to public schools. As I shared earlier, there are few absolutes in American education as it's a matter left up to the states. One clear absolute, though, is the lengths white parents have historically gone to keep their children away from children of color or to try and dictate the terms around which children of color will interact with formal education. We can see this in how literacy for enslaved people was outlawed by their enslavers and state lawmakers unless the lawmaker needed an enslaved person to be literate. The way Indigenous children were treated in Indian Boarding Schools. The ways communities built schools for white children and schools for Black children during the rise of common schools in the North. The way district boundaries were drawn during the era of school consolidation. The reaction of white mothers to the idea their children would possibly sit next to a Black girl. How white parents closed down entire school districts rather than integrate following Brown v. Board. The so-called "white flight" during the 1960s and 1970s. The rise of magnet schools in city districts, which through design and inaction are often whiter than the district-at-large. The way parents advocate for creating splinter schools when a district becomes "too" diverse. In other words, there is another history that shows a pattern of white parents using what power is available to them to continue to keep their children away from children of color. Advocates of this particular history would likely say that the idea of vouchers and charter schools provided the latest tool in the toolbox to continue that legacy.

I am aware of a few books currently being written or about to be published by historians that are trying to build a better, clearer understanding of the history behind the rise of charter schools and vouchers. But, alas, the clearest answer in your question is basically, "because people in positions of power thought they were a good idea."