I think that Benitez is a supporter of whatever club he is managing at a particular time but is not really a big supporter of any club .
You can understand that he was help in high regard by Liverpool supporters, look what he won for them.
Evertonians could not hold HK in any higher esteem and that is because of what he did for the club. But, I don't think he is worried about the legacy he leaves behind or at least would not let that legacy interfere with a job opportunity that may arise.
Benitez is a gun for hire, trying to make whatever club he is managing as good as possible.