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Because doctor comes from the passive doctus ('having been taught', i.e. learned) rather than the active docens ('teaching, instructing').

Not really. Doctor means "teacher" in classical Latin.

Good point. English has both actor (=someone who acts) and agent (=someone who acts) from agere = to act, and both doctor (=someone who teaches) and docent (=someone who teaches) from docere = to teach.

Because physicians are licensed to teach the practice of medicine.

that's the root of the word. The original four-wise distinction seems to have been Doctor of Medicine/Law/Natural Phylosophy/Theology. Just different kinds of knowledgeable people.

Interestingly enough I just realized in Italian there is an old word, "dotto" that literally would mean "someone who has been taught" but concretely is used in the sense of well educated, knowledgeable, wise, cultured.

Bologna, the home of the first university, has the nickname "la Dotta", i.e. the educated one.

For the same reason social science is called science. When words start to become prestigious many will attach that word to what they do.

Social science is science. In what way is it not?

Lack of reproducibility results for one.

That's a problem in all the sciences right now: https://www.nature.com/articles/533452a