The obvious bears repeating. We learn to do something by doing it. If I want to learn to speak Italian then I should speak Italian. If I want to learn to read it I should read it. Cross-training is good, but doesn't provide the skill that direct training does. For example, speaking Italian won't make me as good a reader as reading does.
I've been trying to learn improvisation, and the thing to do there is to improvise. Studying the theory behind is necessary, but you won't learn until you do it. My technique is called Blindfolded Rhythm Changes, and it involves just playing the chord changes of I Got Rhythm over and over again. I've learned some things about it. For example, the improvisational ideas tend to gel and become compositions, and hence on as improvisatory any more. It can get stale, but these stale bits can become vocabulary items or "licks" that make improvisation easier. Once I get too set in a pattern I have to play something else. At some point I will have to learn the changes in another key.
My idea often sound unhip or unjazzlike. This is fine for now, as long as they are intended, heard before they are played, and with some melodic and phrasal shape to them. I'm able to do some improv over my own left-hand walking base, but I feel my ideas are restricted that way, as opposed to having the right hand play what it wants and the left fill in rather than playing four quarter notes. Still, a week or two ago I could play the bass line but couldn't do anything with the other hand.
We learn to write scholarly books and articles by writing them. It would stand to reason that someone who's done more will be better, more masterful. I don't believe in the paradigm of someone who writes and publishes very little and all of it is brilliant. The mass producer of mediocrity does exist, and is a mystery to me.