If a private school is taking public money then they should get treated like a public school
This is an assertion, not an argument.
You could say "All schools that get public funds must have recognized accreditation and meet DoE curriculum and testing guidelines so that taxpayers can be assured that they aren't paying for quackery and kids are getting proper math, science, English, etc."
If schools aren't required to meet these guidelines, then paying them isn't just a waste of taxpayer funds, it's a terrible disservice to those kids, their families, and their communities.
That's a sound argument. I'm looking for a good argument why a school that accepts public funds shouldn't be allowed to deny admission to anybody. Saying "because that's what public schools have to do" isn't even an attempt to clear the bar.