I am very interested in the topics of sightsinging and audiation.
I hope that I have not appeared dogmatic in my opinions. I do not have a clear idea of what is really involved, despite a good bit of discussion and reading. Conventional wisdom that is presented is not very convincing to me.
Here is what I observe:
Many people say singing is essential to learning piano.
Most people I know that can play piano decently are not good at sightsinging.
Many people in vocal choirs, including my daughter and some in our church choir, sing very well but read music poorly if at all. There is more than one path to singing.
Here is what I think I've observed:
I sightsing very well, usually much better than the music majors we hire to sing with our choir.
I have yet to meet a confident vocal sightreader that is not also an instrumentalist. Not saying they can't exist, but in my experience they are rare.
My sightsing has improved with adding more theory knowledge.
My sightsinging is best at familiar genres - tonal SATB hymns are a snap, contemporary syncopated pieces are more challenging, modal pieces throw me until I realize what's going on and practice the mode, atonal modern pieces are really hard.
Audiation is very important to me while playing trombone. Unlike piano, merely having the right fingering does not give the right note. In that respect it is like singing. However the process of audiation does not seem to be quite the same.
My audiation for a monotonic melody within a customary range, say F below the bass clef to 3rd space C within the treble clef is good. (my trombone range extends to the F on top of the treble, but my performances never require anything above the C, I'm not a jazzer.)
My multipart audiation is poor particularly for chords, it's a little better for counterpoint.
What I think, subject to change:
Sightsinging is a good check on audiation, but does not teach it directly. And it is possible to sightsing without audiation, but less common (but some instrumentalists do it).
Multipart audiation is an extension of monotonic audiation but is not learned by sightsinging, and sightsinging is not a good check. (obviously)
Audiation may not be that necessary or even useful to playing piano by reading music; those who play mostly by ear (not me! wish I were better at it) may have a different approach.
Audiation is improved by listening to recordings while following along in a score, and by transcribing.