I think an important caveat is that, at least in American academia, your advisor should not have the same role as a supervisor, depending on the field and research arrangements. As a PhD student, you are expected to have care and motivation for your own research and situation: as sokoloff points out, the idea is that the student pushes and the advisor guides. You can, and in many cases are expected, to disagree with…
Friends in math and political science certainly had experiences like the ones you described, but they would be fairly unusual in a wet-lab field, where students (and postdocs) are largely expected to work on an advisor’s grant. This is especially true when the experiments are expensive: a committee member would need to be generous indeed to put up tens of thousands of dollars for (say) an fMRI study.