thepianoplayer416 There are 2 ways to reproduce a piece of music: by ear or by the score. People who had music training with a teacher would be able to read... especially when the music is Classical.
True. Reproducing music - or at least the 'essence' of the music can certainly be done by listening. Some people can really do it very well - as in generate the gist of it. For accurate note-by-note -- yes indeed, the written score at least allows the original (source) sequence to be properly preserved. These days - where recordings are available (although in many cases, there are no original source recordings as they didn't have the tech back in the oldest days), the combination of score plus recordings has its benefits, because that information conveys even how somebody (eg. the author) played it - in terms of timing, nuances, rubato etc. Timbre and other features might not be exactly the same though of course - as each instrument and environment etc is different.
These days - another combination - is eg. synthesia plus audio recording, which in many selected cases can be effective, especially when combined with audio recording of how the music should sound, and possibly when the students are then also taught intervals, finger sequences (or how to work out workable ones on their own in pieces and/or scales and/or arpegg scales etc), relative pitch training, chord work, keys (ie. key signature) and other elements (eg. the circle of fifth sequences - which actually has short-cut application methods for very practical usage, and can become ingrained into people - the more they just use it, and 2-5-1 etc techniques etc) --- that's if students want to get into these other areas, which can open the door to areas of some musical freedom.
Of course, it's a broad area - but once enough knowledge is accumulated, then people will automatically sense some sort of own musical freedom, and also they can (if they want) learn more and develop more - such as checking out the various other sorts of scales out there, and various music genre, composition counterpoint work etc. There's enough content all together to keep people interested and fascinated in playing piano and making music to last 'forever'.
But back to music score. For some music - where a huge lot is going on, and if somebody wants to be able to view the 'notes' or patterns in a 'most efficient' way (provided that they have been taught to read the symbols relatively effectively and efficiently), then 'traditional' type of notation - music score sheet will really be the most efficient, especially for very 'busy' type music - with a heap going on in it -- lots of notes happening all the time etc.
But when it comes to debate/chats about eg. score versus synthesia etc, I just know that people that get into synthesia can always (or also) get into score reading if they want. What synthesia can do for many people is to get 'foot-in-door'. Get them into piano and music in one way. And from there, they can choose to learn to read music etc if they want.
As for Katherine, as we can see - not knowing how to read score sheets still allows people to generate amazing music of the sorts that they like. She expands/grows in a particular direction (or directions), and there is certainly nothing stopping her from expanding in other directions too if she wants to. But it looks like she has developed adequately and is still developing - and having fun with the piano and music, which is absolutely excellent.