As someone who grew up in the hard-core Religious Right subculture/church, there are a few reasons.
1. Prior to the 1960s, many public schools were very openly and explicitly Protestant. Daily Bible readings, prayers, they basically openly embraced religious belief and practice as long as it was "nondenominational" (e.g., "Protestant"). I remember an older family member complaining about "God being taken out of schools" and saying "well, we had a Jewish person in my class and when it was her turn to read the Bible, she always read from the Old Testament." The Catholic/parochial school system developed as a means of educating Catholic students because the public schools were so openly Protestant.
2. After court rulings banned prayers, Bible readings and open religious instruction in public schools, churches (especially conservative Baptist churches) began opening their own schools. The one my siblings went to opened in 1965. There was a Lutheran school system similar to the Catholic one (albeit on a much smaller scale), but it grew post-1965, too. Some were also created to avoid segregation, but moreso in the South than in Indiana (where, thanks to the Klan's control of the state in the 1920s, most small towns had few/no Black residents, and that's where many of the religious schools were).
3. Parents began desiring their tax dollars that were supporting those "Godless" public schools be able to be used to send their kids to said conservative Baptist schools (or pay for homeschooling, which has also become a big deal in conservative/evangelical culture), and thus the voucher concept was born. Legislators, in trying to funnel money to their preferred private schools, began to villify public schools, and the distaste has grown since.
In Indiana, every single bill post-2009 has been to try to weaken traditional public schools and funnel as many kids as possible to charter/private/parochial schools. And when not enough kids are doing what the legislature told them to do, they double down even more.