Private schools can use the games, they just have to be not-for-profit (as opposed to a for-profit school, like DeVry). The vast majority of private schools are not-for-profit.
But yeah, an entity's ability to draw a net positive profit doesn't change their for-profit status, obviously. As for why Zachtronics is limiting this to non-profit schools, my guess would be for tax purposes. There's a good chance there's a benefit from providing something like this for non-profit institutions, but not for for-profit ones, although it's been 7 years since law school for me and my tax law is rusty.
Although there's always the chance that he just didn't know for-profit grade schools exist, because I sure as heck didn't.
"Private and/or for-profit schools are not included. However, if you email us with the information listed above we can put together a quote that includes an educational discount (usually 25% - 75% off depending on the number of licenses)."
We didn't take any charter money because it comes with strings. We were doing our own curriculum development. (Everyone there had a healthy distaste for the standard curriculum that gets taught in most places.)
And before you start, we were secular. Most of us, including myself, were/are atheists!
Maybe I did work at an edge case, but I still think that this is an edge case the world needs more of.