We do not know Kaori's progress in her violin studies, and as I am not a violinist, I cannot comment on the difficulty of her repertoire (and thus, of the competitions in which she plays). However, there are two out-of-universe reasons for Kaori's lack of fame that come to mind.
Firstly, even classical music listeners might not follow competitions closely enough to recognise those who performed well in them. I mostly listen to classical music and I know the names of a number of big competitions, but I barely know the names of those who did well. Moreover, with the exception of certain "niche" interests — period instruments, transcriptions, some types of vocal performance — I am likely to end up listening to recordings and performances by better known, more "established" musicians whom I already know.
Note also that there are any number of decent musicians who play difficult repertoire but are not well-known outside of their local circles.
Secondly, it is made clear by episode 2 that Kaori does a poor job at following the score, which might prevent her from getting a following in the right circles.1 I might avoid someone whom I knew to consistently and flagrantly push an interpretation that clearly was not the composer's intention, and I am not the only one. My teachers would also be unhappy with me if I played like that, which in turn would inhibit my development and recognition.
- When I speak of "following the score," I do not mean robotically playing it without thought. Rather, I mean that the production is intelligent and approximates what the composer might have expected. It is important that I "stick to the score" in my piano playing, but when I do so, I still have to make decisions and play thoughtfully.