Graduation descends from Latin gradus, step or degree, through medieval Latin graduatio, the conferral of an academic degree, with graduates literally stepping up to receive the degree. Gradual shares the gradus root. Indoctrination descends from doctrina, teaching, from docere, to teach. The two roots are unrelated, and no historical source combines them.
The gradual-indoctrination story circulates as a gotcha about education: they even hid it in the word. It spread because it flatters two audiences at once, cynics about schooling and people who enjoy hidden-meaning reveals. Neither audience checked a Latin dictionary.
The honest version of the skepticism does not need the fake etymology: education systems genuinely did standardize credentialing in steps, and critics from Illich onward have argued schooling sorts more than it teaches. The argument stands on Deschooling Society, not on a word game.
The false etymology survives because it feels true, which is exactly the mechanism this archive exists to study. What other explanations are we believing for the same reason?