The role of a tutor is to find your weakest point (or points) and give you personalized advice on how to improve on them. It is important that you can put trust on the tutor, as your weak point is likely due to a blind spot. When given criticism, it can be viewed as objective or subjective, i.e. a "question of taste", and thus the criticism would be viewed as invalid and not acted upon. As to whether the tutor should point your weakest point or some combination of weak point, it should depend on what helps you learn the most efficiently; it might be to focus on one aspect, or work in a more integrated fashion.
Last (although that may be first), they should consider what are really your learning objectives to tailor that advice.
Definitely not just grading. Tutoring is explaining and back and forth discussion to impart knowledge, in context and in response to specific difficulties/confusion the student is having.