Some actual numbers. I live in a suburb of Seattle and have three kids, two in private school and one in the local public schools, which are excellent. I have two in private school because before moving, the local public schools weren't so great. They'll transition to public in middle school.
1.) We pay about $12,000 in property taxes annually. Our house is worth close to the median sales price in our zipcode, maybe 5-10% more.
2.) Our local school district spends $14,000 per student and has revenue of $19,000 per student. According to US News, 60% of school district spending are related to instruction, 35% to "support services" and the rest to other. It's unclear where retirement costs for retired teachers and admins comes in.
3.) The local school district where our private school is located has revenue of $23,700 per student.
4.) We pay $24,000 for tuition at our private school. We love this school, and in my opinion, it's better than the local public school.
From my personal observations, there are three major things driving education costs:
1.) Explosion of admin and other non-teaching positions
2.) Retirement costs for all employees. Here in WA, teachers can retire very young and get a pension. Many or most will be compensated more after they stop working than while they're working.
3.) Catering to special needs kids. A special needs kid by state law is guaranteed to have their educational needs met. For many very needy kids, this means a one-on-one helper during the day. So the rest of the kids are in a classroom with a ratio or 25 or 30 students to one teacher, while our special needs kid has 1:1. Without the special needs kid, you could literally halve the size of the class. Private schools don't have to and usually don't accept kids that would require one-on-one care during the day. Many other special needs kids don't necessarily need one-on-one assistance all day, but might need one-on-one care for an hour a day, or three or four to one attention, again drastically reducing the resources that can be spent on the other kids. This is a huge issue in poorer school districts where kids coming from poverty or homes with more screens than books would really benefit from that 13:1 student teacher ratio, and instead the resources a getting sucked away to benefit needy kids with assertive parents.
The student teacher ratio at my daughter's school is 19:1. Her classes have 30 (or more) kids in them. Why? Special needs kids.