Gadol > doesn’t want to have to monitor beliefs/teaching of religious schools
why should they? State is funding general education that they can monitor by testing. If the bochrim can’t write or count, the subsidy goes away. As I suggested above for people, you can do this for schools – every school (including public?) is reimbursed after a test. As of now, some Jewish school participate in private-school tests, but do not make test results available even to parents, at least in my experience.