Sure, but if you are a private school charging $20k a year now, why wouldn't you charge $25k when every potential student gets a $5k voucher? Then we need to increase the size of the voucher to meet the original goals of the program. Rinse. Repeat.
Parents obviously want to minimize the amount they're paying out of pocket, and they'd like to keep some of the voucher money, so high-end schools as well as low-end schools are incentivized to keep their costs down.
It's true that many families will pay more to buy a house in a good school district, but I suspect that the school itself isn't the whole story: rather, many people interpret the school district's reputation as a proxy for the safety of the neighborhood and socio-economic status of its residents, as well as a proxy for their own socio-economic status.
Ulimately, I suspect that with the free market voucher system I described in my parent comment, legacy private schools would continue to charge outrageous tuition. But I think we'd also see some new, possibly virtual "bang for your buck" schools which would compete on price and which would be exclusive due to entrance exams and high academic expectations for enrolled students.
[1] https://www.edweek.org/leadership/education-statistics-facts...
This spherical cow is fully confirmed by the higher ed market /s.