In other words, how deeply connected with music do you have to be to be a virtuoso? Is mechanistic perfection sufficient or do you have to have proficiency in all aspects of music, imrovisation, composition?
If you want to be an improvisation virtuoso, you need to practice improvisation.
If you want to be an composition virtuoso, you need to practice composition.
If you want to be an X virtuoso, you need to practice X.
One could also make the case that learning X can be (more or less generalized to) Y. It would be really interesting to see studies of skill in musical proficiencies and how they may/may not correlate with each other. If someone knows this type of research, please chime in.
A virtuoso is one who is highly skilled in technique.
Technique is mechanical skill in an art. Or a manner of artistic execution in music.
Improvising obviously needs mechanical skill in playing an instrument, otherwise even brilliant ideas won't come out right when played. However, the second definition seems to contradict that - what 'manner' is subjective.
A virtuoso is often seen as a wonderful player. Someone who can deliver a concerto or suchlike impeccably. That player may only know that one piece, and can only play it verbatim the same every time. Unusual, true, but possible. Maybe couldn't improvise to save his life. That's a completely different skill, so the question is probably unanswerable. But I tried...
Short version: they all inform each other, but are not prerequisite.
I'm not a professional, but I am reasonably accomplished in improvisation, composition (my university major) and playing memorized piano pieces.
I started improvising because running scale patterns and doing arpeggios all day was too boring-- so I'd run up and down scales, throw in some arpeggios, practice modulating from key to key, and so on.
Having learned theory makes it MUCH easier to learn new pieces, especially classical ones which follow quite predictable harmonic motions.
Playing piano makes it MUCH easier to learn to compose, because you can play all 4 parts of a traditional SATB harmony yourself.
Very many great pianists have one or two really top-notch pieces that they've written for their live performances despite not really being big-C "composers"
And of course, pretty much every great composer since Bach has been an expert keyboardist, with a couple exceptions.
This question is very convoluted and in my opinion self contradictory.
"Do you need to improvise to be a virtuoso?"
"how deeply connected with music do you have to be to be a virtuoso"
"Is mechanistic perfection sufficient or do you have to have proficiency in all aspects of music, imrovisation, composition?"
How are these connected in your mind?
One could argue that you do not need any mechanistic perfection to improv. So why would one think improv is a criterion for achieving virtuosity?
The answer would depend on the type of music one is discussing. In the classical world one may achieve virtuoso status by playing straight (no improv). And I suppose a certain degree of mechanistic perfection is required to perform high level pieces without mistakes and glitches. But at the end of the day it's the musicality that one listens to. Even the greatest musicians have squeaks and scrapes in their playing but are still considered a virtuoso.
In some genres of music one learns only by improv. Mechanical proficiency develops as one needs it. Within those genres people would be considered a virtuoso by improv skills alone and not merely mechanical proficiency.
There is too little to go on. But one thing I'd say is that mechanics alone does not make one musical.