This is significant to me, as I suspect many of the people with anxiety that is significant enough to be diagnosable would not be willing to do it or would pretty much shut down if placed in the setting.
To delve into personal anecdotes:
I had a theater class in school that regularly included improv exercises like described. In 11 years of it (K-10 or so), I don't think it ever went well.
The actual manifestation of the problem with "no script" was drawing a complete blank as soon as I was asked, not having a bunch of ideas I was scared to respond with.
It's certainly possible that these people have a better method, but I had multiple theater teachers and I was socially fine in normal settings. All these activities tended to do was give me more anxiety issues because everyone else could come up with things and I'd just be stuck there no matter how people tried to help me get going.
Those are not really common situations in adult life, but I'm still bad with "icebreaker" games and other similar things that "put me on the spot" to come up with something outside of normal social interactions.