Would you call the character of Doctor Harry flat or round? Why is his flatness or roundness appropriate to the story?
I would say that Doctor Harry is a flat character, and that his flatness is appropriate to the story. He's flat because he's just a set of tics and external characteristics, described from the outside, and because he exhibits no change or emotional complexity.
It is appropriate because this is Granny's story; he's essentially a background prop in her psychodrama.