Correct. Schooling may not occur in a school. The only schooling done in Public schools is indoctrination. The word is both a noun and a verb.
Yes, you are of course right about the different uses of the word ‘school’.
But your original piece, with “One must be taught to think”, implies something other than self-teaching. Your further comment clarifies your thinking, if you will.
Again, I believe we are in agreement.