Who says those people are right, and they are not outright hypocritical?
We live in a militaristic, capitalist culture. Rehearsing for competition in the job market is half of our education. Before that it was competition to be seen as valuable in war so as to belong to the army where you are most likely to simply survive. Reputation is vastly important to the whole of most human cultures throughout their histories.
People state the opposite as an anodyne, not a disavowal, or they wouldn't bother saying wise things at all -- they would keep them to themselves. Reputation is the only reason people take your advice, if it really isn't important, why put in effort that should not really have any effect?
From a pacifist POV like Starhawk's, a focus on reputation is actually a major step forward from assuming that force is the appropriate way of deciding who listens to whom. To the degree we can establish reputation as a real force independent of bullying and hegemonic control, we are that much closer to a peaceful society where 'power over', 'power with' and 'power within' are properly balanced, and not skewed by the environment.
More absolute approaches to pacifism are just sexist. They blame men for doing what was historically necessary for their survival, and raise women up on a pedestal as the more peaceful sex, even though that was not really a choice, and is not really a gain. If the answer is balance and not hegemony, it is important to defend the things you want to balance. In a capitalist culture, force praises itself, and we have equally many praising internal development, but the point in the middle is also important. Having places to practice the skills involved is useful.
In the meantime, major projects like #MeToo make real use of reputation as a tool to balance power. And it would be unfortunate to undermine them with the idea that their main tool is really an illusion.