where you defer to someone else for their expertise because maybe they’re the only doctor available who can treat your illness, so you need to do as they say to get better.

you have the right word for it: expertise (see my other comment).

it becomes a hierarchy if the doctor involuntarily hospitalises you or uses the courts to force you to undergo the treatment; the power (force) to do that is authority. so long as you still have the power to challenge or otherwise discuss the prognosis, it is not a hierarchy, especially if the treatment is gratis and libre.