Interesting. I had a similar experience mentoring someone who had a lot of potential but tended to stutter when presenting to clients until they would get in to a flow. What I did was gently say that "I, too, am sometimes nervous presenting to a room" of strangers (usually executives) during project pitches due to the stakes.

So I told them that my trick was to get them talking about their business straight out of the gate (because most people like talking about themselves at that level) and their stutter went away.

I feel like that's maybe more constructive approach.