A private school is a business that offers an exclusive educational experience. It's similar to the difference between free golf lessons at a public course and lessons from the golf pro at a private country club. They are not the same, and the latter should not be funded with taxpayer dollars.
Private schools provide a specialized environment for families who value that experience and are willing to pay for it. There is nothing wrong with that. In fact, many private schools have successfully operated this way for hundreds of years by competing in the marketplace and delivering a product that families choose to purchase.
If a private school provides a quality experience, its success and survival should depend on the families who voluntarily choose it—not on taxpayers. No taxpayer should be required to subsidize another family's exclusive educational choice.