Disclosure: I am not an accountant or tax lawyer.
In the case of school districts, easily the largest line item of the SALT liability for New Yorkers, The quo would be educational services. Anyone who donates to one of these "charities" and then puts a child in the school system could be at serious risk for tax evasion. It doesn't matter legally if the "charity" is also floated by other revenue streams. It's about what the filer claims and what he receives. I am pretty sure the IRS has a history of formulas to apply for figuring out how much is rendered in services[0]. Per pupil expenditure would be a likely candidate. Perhaps those with no children in the system would be safe.
The analogy with the church would be if they were giving your child a free or discounted on religious school admission as a contingency of how much money you donate. You can't write off donations like that[1]
[0] https://www.irs.gov/pub/irs-pdf/f8283.pdf
[1] https://www.irs.gov/publications/p526#en_US_2017_publink1000... - "Contributions From Which You Benefit": .... "Tuition, or amounts you pay instead of tuition. You can't deduct as a charitable contribution amounts you pay as tuition even if you pay them for children to attend parochial schools or qualifying nonprofit daycare centers. You also can't deduct any fixed amount you must pay in addition to, or instead of, tuition to enroll in a private school, even if it is designated as a "donation."