My father spent his career establishing and running large research laboratories, and he taught me early on that a manager’s job is not to “manage” her/his people. A good manager’s job is to hire the best people he/she can find, and then protect them from outside interference so that they can get on with the job at hand.
My principle has always been that a manager’s primary job is to run interference for his/her staff, and to do whatever tasks need to be done, no matter how “menial”, to help her/his staff get on with their jobs unimpeded. There are some other important tasks for managers – mentoring young staff is one, helping to provide a guiding vision is another, encouraging staff to improve their skills and keep up with their field is another. But the single most important task is to face outward and protect one’s staff from the inevitable inanities of the corporate bureaucracy that always exists.