I suspect it very much is a practical skill. I'd be shocked if genetics are the sole determinant in your wit - surely other aspects such as linguistic ability, social skills, knowledge of pop culture, confidence, etc. all contribute to wit. And those can all be trained.
It would take time, of course, and a lot of practice rather than just having it explained to you. But I don't see why it wouldn't be trainable.
I thought about learning to play violin when reading this article, a teacher might explain you everything but it's impossible to understand as it's something so unnatural and driven by feel and experience. In fact you start playing with not great technique for a few years until you develop the ability to start working on the finer details and so on
I mean, obviously you can explain many things to someone who can't replicate it "just in time", riding a bike is one of those examples.
Explain math, playing guitar, reading, or whatever perfectly to a person, they still have to practice to be able to do it.
The problem isn't the explanation, it's the difficulty of the skill to learn.
Don't know why we need a new word/phrase for it.