I find it interesting that I don’t stutter when I read, even if I’m reading in front of an audience. Recent research has discovered that some people who stutter may have a social anxiety which is contributing to the stutter.
It would seem to me that the act of reading is different than actually speaking your thoughts. Perhaps different parts of the brain are being triggered for one task versus the other. This in turn might be why someone might stutter at work but then never stutter if reading a story to their children.
Thinking back to The King’s Speech movie, the King actually did stutter when reading a story with his children. Stuttering is very unique to each of us so perhaps it is true for a percentage of people who stutter that they stutter in all speaking situations, and show no differentiation between reading or speaking.