Is “Be Yourself” a terrible advice for a leader?

I was at a coaching session with a senior manager of a services company. We were discussing the challenges that she was facing with some specific team leaders. At one point, when discussing solutions and action steps, she asked me “Should I just be myself or try this instead?” She was thinking of having a different approach which would not have been her natural response. The question struck me because, on the one hand, I believe in authentic leadership. I believe that leaders should naturally be interested in performance through people. They should naturally be assertive and offer constructive feedback and appreciation, and naturally communicate the purpose or the mission. Leaders should be natural and authentic at all times because their role is not linked to a schedule (ex: from 9 am to 5 pm). It is a 24/7 role where people are watching your behaviors and words, and are translating them into things to consider (follow) or to criticize.