I totally agree Côte, it's not the same level at all - the point was more about the idea that you can learn to improvise, it's just that most classical students don't (presumably because exam curricula do not impose improvisation as a skill that should be measured, and most teachers aren't in the habit of teaching it).
I think what happens to many good musicians is that they spend the period when learning is easiest (i.e. childhood) concentrating on technique and interpreting other people's music, and that the route to improvisation gradually gets blocked off by fear of failure.
I also think theory is often badly taught (as an exam passing exercise, or - as is the case here in France - primarily as a means of training orchestral players who need to be able to read notes and rhythms well). DS has a far greater understanding of the building blocks of music than DD or I have.
All the musicians I know who are really good at improvising are either self-taught or have done some jazz classes, and for most it was a fundamental part of their musicianship right from the start. It seems to be quite hard to "tack on" improvising skills at a later date.