This is getting to be a pretty interesting topic and the replies are fun to read and respond to, so I'll continue my thoughts.
Also, I think most people posting here are improvisers because the post kind of naturally resonates better with those who have a predilection towards composition. Many of the frequent posters in the forum improvise as well (which is rather unusual).
This struck me a bit. I think it's because those with an inclination to improvisation/composition are generally "deeper thinkers", for lack of a better term. Not that classical performance isn't "deep" per se, but I think one can get away with giving "perfect" performances (provided they have practiced the piece a lot and wisely) without thinking or feeling too much, whereas improvisation and composition in general, more often than not, probably necessitates "thinking/feeling" more.
And people who think and feel are probably more likely to discuss these thoughts/feelings.
There is a philosophy practised by some learned musicians that nearly all of the authority of music creation originates from the composer, and that the performer's job is not to be creative but rather to recreate the intentions of the composer. Additionally, there are those who take it a step further and profess: that in order to recreate the intentions of the composer you also need to replicate the conditions in which the composer wrote the music.
What does the performer want for his audience? It's all about that I think. The "accuracy" movement is probably more about turning classical performance into a "documentary", making the listeners know "how it was like". Whereas a more "liberal" approach would mostly be about how the performer thinks the music should sound like, and that's it.
This is a good illustration of the "dilemma" of classical interpretation I think. How far do we want to get in "accuracy" of interpretation? What even does "interpretation" mean?
This gray area, this "forest", I believe, is where creativity in classical performance lies. It relies on the performer's understanding of the music, and also the audience's understanding of it. What is the audience looking for? Who are they - are they relatives of the performer? Panelists? Competition judges? Non-pianists? Etc.
It's a little like a speech. The credentials of the performer may even come into play. Maybe what he's wearing has an effect as well, as well as the idea that the audience perceives his demeanor or attire to project. The setting also has a role. Does it affect the acoustics? What type of piano is the performer using, and does it matter? How expensive is the concert, and for what purpose is the performer playing (charity? competition? self-fulfillment? advocacy of a certain composer or musical style? etc.)?
So, even if you're playing a set of pre-written notes, there's still so much to consider, so much to explore and work around. As someone who lives in a place where classical music is not "native", there's hardly a market here. I grew up with an inclination to common practice classical music, so my appreciation of it is something that many in my culture probably won't understand. This realisation frustrated me a bunch of times in the past, but it's also a challenge for me and a big question that I've set myself to work with for some time.
So "replicating the conditions" I imagine, won't do concerts in my environment a favour, because they don't know much about the history of this kind of music. It would probably be much easier to do it in Europe, I'd imagine, due to the pre-existing familiarity with the context of the music there.
What I find perplexing is how little improvisation is mentioned in these historical context studies in performance. The musicians of the day were more concerned with just creating music, and improvisation was an integral part of that.
I think there are a few out there who tried improvising as well, like Gulda for instance, and I remember there was this wonderful recording of a period rendition of the Liszt Sonata with a prelude in the beginning.
"Just creating music". Indeed. That's what's strange about the whole classical music thing. It's hard to make people today experience the feel of the music because we didn't grow up in the circumstances of that music. We can get closer to it by reading about the composers of course, but that would require more effort.
Here's a simple practical answer: how many legit record companies would think they could sell improvisations to the public.
Classical music fans probably want their favorite piece listened to. They're interested in the music they already know and they want to hear it. When you improvise, you're the composer, so by selling records of your improvisations it's practically the same as selling records of your compositions.