for hospitals, colleges, preschools, and universities to obtain public money, they are required to meet a number of regulations. And they all do. 
Even now, charter schools are private schools that meet public regulations, and therefore are funded. 
It is only in k-12 that you see private schools fighting the regulations while at the others, they go along with them even though they are as strict. 

So, the issue is NOT on the public side, but on the private side. 
Sadly, the private K-12 schools are more about indoctrination than about helping kids. 
Oddly, even a number of charter schools skirt this issue and push Young Republicans (other wise a redo of Hitler Youth), as well as bring in loads of GOP politicians, but NEVER 3rd parties or Dems. IOW, for many of them, it is POLITICAL indoctrination, not religious. Sad. And yes, this include AA. We just make sure to talk to our sons and prove the lies that GOP say there.