When searching for a private voice teacher, the presently accepted and presumed expectation for potential students (and their parents if younger than college age) is that it will be a one-stop-shopping affair, including the study of vocal technique, a variety of different musical styles, repertoire and language coaching, dramatic interpretation and expression and the added benefit of a built-in, song learning accompanist.
As a voice teacher with over twenty years of experience, I have engaged in all of the above, yet I find I am ultimately a better teacher when I step away from the piano and teach on my feet, walking around my students, observing their posture and alignment and and giving me the ability to look into their mouths to see if they are holding tongue tension there. I also move around with my students (yes I ask my students to move when they sing, especially when then are physically locked and they are having a tough time feeling the rhythm of the music).
My students are often asked to sing a capella instead of my spoon-feeding their every note. It amazes me that many singers cannot sing a descending major scale in tune by themselves. This is often one of the first thing I insist they learn to do without the assistance of the piano. Not only do my students sing better in tune with themselves, they learn to be independent of the piano, accompaniment tract or whatever else they have come to rely on.
The University where I teach has a wonderful student accompanist program which acts as training ground for building collaborative skills, especially for our piano majors. I think this program makes a lot of sense for these students because, in addition to teaching piano lessons, a large part of their professional lives will be spent accompanying singers and ensembles.
Coaching student accompanists/collaborative pianists and singers in voice lessons is definitely one of my passions. My voice students learn to interact, communicate, and lead with their partners. The pianists learn to listen, breath, balance and sync with the singers they play with.
The beneficial result of this teaching model is that my voice students have my full attention in lessons rather than the distraction that potentially results from my reading and playing music at the keyboard while attempting to listen and pay attention to their singing.
While I respect and admire my vocal colleagues who are able to effectively multi-task in this way, for me, stepping away from the piano makes me a much better teacher on many levels. One-stop-shopping may not always be the answer.
Here is a link to a blog that describes the difference between a voice teacher and vocal coach which echoes what I've said above;