Unfortunately, things are not as simple as they seem!
Before venturing into difficult matters, let's have some simple discussions (!!!!).
What does improvising mean?
Does it mean solving a problem with what one has at their disposal?
Does it mean doing things that appear original in very little time?
Does it mean creating forms/structures/responses/theories that no one has ever imagined before?
In fact, no one has yet clearly understood what improvising means, nor even composing (and, I say, not even performing!).
Therefore, I believe that the discussion we could have regarding this risks leading nowhere, because each of us gives different meanings to the same words.
Moving to an even more difficult and intricate level, let us remember that we move according to conceptual schemes that are largely the product of a rationalistic and deterministic culture. This, on one hand, has represented the most significant advantage of Western countries in the economic struggle but, on the other hand, is also responsible for cultural flattening, unless it is appropriately mitigated by sensitivity and attention to diversity.
For example, we say that written Western music is art music. Perhaps the other is uncultured?
Furthermore, we define popular music as a certain type, based on its apparent simplicity (rhythmic, harmonic, melodic). Unfortunately, in doing this, we completely ignore the enormous value of cultural transmission that it has had and quite a few other things; moreover, let's be honest, we assume a somewhat (?) snobbish attitude.
This reminder of mine is not accidental.
For example, PianoScholar claims not to be attracted to Jazz (they can do it very well, because it's a matter of taste) "also because you recognize it immediately by that unmistakable particular flavor" (I can't accept this! If I adopt this criterion, I must throw away Mozart, Chopin, Bach, etc....., "because you recognize it immediately by that unmistakable particular flavor"!).
In fact, even improvisation has rules or patterns, depending on how more or less hidden the genres are; however, the best improvisations try to use only a few very simple patterns, but with a great capacity, with few uses or iterations, to "land in a new world".
This world can be rhythmic, melodic, or even of a completely different type, as happens, for example, in Hindustani music.
In the end, improvising may not be different from composing, even if we give the term composition a typically "modern" meaning.
It is no coincidence that Camy86 reminds us that the improvisation that "everyone" continuously performs is in the construction of language, to communicate concepts.
Some write them down and they are beautiful, and they are poems or musical scores. Some read/perform them and they are interpretations.
To return finally to the initial question, I believe that Jazz is little practiced because doing it well is difficult (like performing classical music well, perhaps even more so) and, if you don't believe me, try having a go at Oscar Peterson's book.
But it is above all difficult because, unlike interpretation, the dimension of improvisation (of quality!) is more subtle and imperceptible than that of interpretation, as it is closer to creativity and further from introspection.
Finally, I recommend a pleasant reading:
Carlo Serra, On Improvisation http://users.unimi.i...m1/dm1impcs.htm