Do movie actors have to know about script writing? about fliming angles and the structure of the plot? of course they do. Do they have to writers by their selves or be able to improvise text and write interesting stories by themselves? of course not! they should only know what they need in order to perform the most convincing and meaningful act.
Same goes for classical piansts. We have to know what we are playing we have to know the theory, and try to analyze and understand the music the best we can. for example we should know that we are playing a dominant, and feel that it is "going to" the tonic, because this understanding helps us to put more meaning to our playing. but we don't have to be able to improvise.
As for me, When I do improvise, I don't trouble myself with thinking about chords and scales, Although I know the theory. I usualy play "whites only" or "free style" (which could sound auful I admit) but this how classical musicians think, we don't like to think so much when we're paying. we just want to play and feel the music.
lol..funny, it still makes me happy.
ok, that was rude. not really a valid "actor" analogy. because I play jazz and you play classical (which I do as well, btw

does not make you and "actor" and me a script writer--we are both actors--but your "acting" is serious only, whereas I can easily do either "serious" or "comedy"--and I earn a living playing both. I can step into your world my very young friend, but you are not yet capable of surviving in mine.
and btw as far as "not having to think" ---I am MUCH freer to "feel" the music and "not think" in Jazz. I have enough exp. in both genres to know that. You cannot yet say whether jazz is "free" or not because you can't play it-- and that, is what got me through music school. sorry, I still have a bit of music school neurosis, after all these years. I don't mean to be so harsh, but classical players drive me crazy trying to tell me how they think it "feels" to play jazz. I have a student with a masters in performance trying desperately to learn to improvise--and paying me to teach her. she's not the first or only classically trained pianist I've met who didn't SERIOUSLY regret not learning how to improvise.
and you are on the right track in learning to improvise with your "white key" thing that you think sounds awful--it's your classically trained brain telling you that it sounds bad, not your chops.

best of luck to you in your musical pursuits.
