I love Mugi because she is the embodiment of what every human being should strive to be. She is heavily inspired by the character of Father Zosima of the Karamazov Brothers, and Dostoyevsky's theme of living through harmony by accepting commonality with the ordinary folk explored throughout the novel, as well as showing love for god through unyelding love for all his creation.
Mugi is raised as an aristocratic figure in a rich family, but feels unfulfilled by this and is drawn to the close relationship she saw between Mio and Ritsu, common folk struggling to keep a simple club together but finding joy in each other none the less. She is affected so much by her simple life in the music club that despite seeming being a mere observer, she is the first to emotionally break down upon realisation that it's all going to end soon in episode 24.
"I've been struck all my life in our great people by their dignity, their true and seemly dignity. I've seen it myself, I can testify to it, I've seen it and marvelled at it, I've seen it in spite of the degraded sins and poverty-stricken appearance of our peasantry. They are not servile, and even after two centuries of serfdom they are free in manner and bearing, yet without insolence, and not revengeful and not envious. "You are rich and noble, you are clever and talented, well, be so, God bless you. I respect you, but I know that I too am a man. By the very fact that I respect you without envy I prove my dignity as a man.
In truth if they don't say this (for they don't know how to say this yet), that is how they act. I have seen it myself, I have known it myself, and, would you believe it, the poorer our Russian peasant is, the more noticeable is that serene goodness, for the rich among them are for the most part corrupted already, and much of that is due to our carelessness and indifference."