Average spent per student is a potentially misleading statistic. The bulk of the middle require little more than grade-level curriculum and materials. But special-needs students and English-language learners can easily require order of magnitude more resources than the "average" student. These are the kids that regular private schools won't deal with but public schools are required to accommodate.
If a non-verbal kid in a wheelchair rolls up to the local parochial school, they may not overtly discriminate, but they'll do their best to convince the parents that their school is simply unable to meet the family's unique circumstances at that time. When they then enroll their kid in the local public school and establish an IEP, there'll be a full-time paraeducator accompanying the child, plus (at least) weekly sessions with physical therapists, occupational therapists, and speech pathologists - for starters. The district might even provide some in-home learning support, depending in part on whether they've ever been sued for failing to meet their ADA obligations. Needless to say, the cost to the district for that kid will blow up the "average" per-student cost.
Someone made a comment above about if there was more competition, there would be a private school for everyone. There already is - you just have to have $50,000 a year to pay for it. There was a daycare we looked at for our son that had a boutique K-8 school attached. They marketed themselves for highly individualized attention for severely autistic children (the K-8 portion, the daycare was just a daycare). It was one kid per classroom with three or four full-time employees per kid. How many families can afford that level of intensive services? Those that can't have no choice but to rely on the local public schools, who are required by law to accommodate those students and provide the resources necessary for them to successfully learn to the best of their ability.
Private and charter schools aren't automatically better. Many charter schools have failed entirely, sometimes leaving families without a schooling option in the middle of the year. Most others don't show any better results than local public schools. The few models that do show great success require long hours, significant buy-in from parents, and strict adherence to rules by the families, something that many (most?) families cannot manage.
There's no magic bullet for improving education. Money isn't a panacea, but there are areas where a small amount would make a great difference. I see it on the opposite end of the spectrum, with kids designated "gifted" who are left bored to tears at the pace of the regular classroom. The kid who are two grade levels behind in reading have aides come in and work with them, but there's nothing analogous for the kids who are two years above grade level.