You are right.
The gubbamint has no business running schools.
If people wish to voluntarily contribute funds to form schools for the propagation of opinions which they deem important enough to inculcate their children with, so be it.
But don't hold a gun to my head and say it is my duty as a citizen to contribute.
Jefferson did not bifurcate religious from other opinions. Opinions are just that - opinions.
My exact feelings about public schools.