"A cheaper house in the lawn-ornamented suburbs where public schools are good."

They ain't no such thing. The value of the public school vs private school tuition is factored into the price of the house. Invisible hand of the market, etc. That's one reason so many folks are now caught in mortgages they can't afford; just trying to get their kids the advantages they never had, but find themselves locked in whereas private school tuition can be discontinued if necessary.

BTW, where do you get a good private school for $9,000?

She told you: Kansas City.

Here in "flyover country" you can go to secular Pembroke Hill, Catholic Rockhurst or St Pious X, or even Montessori for reasonable prices.

Personally, I'd rather homeschool through high school and at 15 have them enroll in the local community college's program for high achieving teenagers.