How does supervisor contribute to students' papers
This depends enormously on the field. In mine, mathematics, I never, ever claim co-authorship on my students' published papers arising from their PhD theses, nor from papers immediately afterward. I do expect to use my expertise, such as it may be, to help students identify feasible-yet-worthwhile projects, and to help them understand both the feasibility and worthwhile-ness, and this can continue beyond the PhD.

(My model for this behavior, which many people disagree with, or find "unethical", or find "degrading-to-students", is that the advisor's responsibility is well described by master-to-apprentice in an archaic sense. Or, similarly, parent-to-child in a more modern sense (since in antiquity children were not quite as valued as nowadays, apparently). I do understand that many people enjoy the idea that students are junior peers, etc., but, in fact, I claim that this possibly-seemingly-respectful attitude is almost always used as a rationalization for abrogation of responsibility... rather than the positive senses one might hope for.)

I do understand that in some fields advisors are invariably co-authors, even if their contribution is no more than funding.

The point is that, given this range of actualities, one can only look at the conventions within one's specialty, because it is simply not possible to predict that expectation from an abstract quasi-academic notion of "who should be a co-author".