"with regard to organ playing, there is nothing to it. You simply strike the right notes at the right time and the instrument plays itself."
--J. S. Bach
When it comes to accompanying, it is necessary to multi-task. Of course, you have to be able to turn pages and play simultaneously, often finding clever ways to play the entire passage with one hand while doing so. But more to the point you have to be able to listen to your choir (or soloist) as well as listening to yourself.
As soon as somebody needs help, you've got to be able to provide it. In the course of a standard choir rehearsal, I almost never play the written out accompaniment. This is because generally I am helping one section or another with their notes. I may be playing the voice parts instead of the accompaniment, or some combination of each.
But even if the choir is singing alone, without the aid of the piano, I may step in to help at any time. We encourage the choir to do as much singing alone as possible, even in places where the piano would be there for them in the performance of the music. This is to help strengthen their sense of their own notes, and so that the director and I can listen more carefully to the sound they are making. The point, after all, is to make sure they can do it. They are mostly amateurs, and they don't spend several hours every day practicing. But they can sound quite good when they work at it.
As they are singing "a cappella," I am following along in the score, and, at a nod from the director, or a whispered "help the tenors" (by which point I am probably already poised to do that because I can also hear that they aren't finding their notes) suddenly out pop a few piano notes. I don't play their entire line--they don't need it, and I am not there to hand-hold, just to offer aid when necessary. That may mean I only play a few notes every 30 seconds or so. But they have to be the ones that are needed. The right ones, at the right time.
There are also situations in which I leave out notes. If the accompaniment features an occasional clash between the piano and what the sopranos are singing, during an early rehearsal I may leave those notes out in order not to lead the sopranos astray, because they will think they are supposed to match the piano. Once they know their part well, I'll put those notes back in. This requires me to understand the relationships of the notes at a glance, to categorize not only the important notes in a phrase, but to see the notes that will be helpful (if the composer has written the baritone note in the accompaniment a beat before they come in, for instance) from those that will not be.
I didn't major in accompaniment at the conservatory, and only took a class or two, so I don't know if this is ever taught in music school, but it is certainly important. And it is a case when it helps to be able to improvise and score read, enabling one to play a few beats of accompaniment, a beat or two of some voice combination, back to the accompaniment, just the sopranos---the situation dictates it, what one hears, and no preconceived plan. It is all done in reaction to where the group is, and what they need. In concert as well as in rehearsal.
"....If you could see him...not only...singing with one voice and playing his own parts, but watching over everything and bringing back to the rhythm and the beat, out of thirty or forty musicians the one with a nod, another by tapping with his foot, the third with a warning finger, giving the right note to one from the top of this voice, to another from the bottom, and to a third from the middle of it--all alone, in the midst of the greatest din made by all the participants, and, although he is executing the most difficult parts himself, noticing at once whenever and wherever a mistake occurs, holding everyone together, taking precautions everywhere, and repairing any unsteadiness, full of rhythm in every part of his body--this one man taking in all these harmonies with his keen ear and emitting with his voice alone the tone of all the voices..."
---Johann Matthias Gesner on J. S. Bach in rehearsal!