Nobody told me when I started teaching voice that there's a lot more to it than just teaching someone how to sing. I am at times a personal trainer. Other times, a psychologist. A cheerleader. And often times, someone to talk to. For some of my lessons, the actual voice instruction is a secondary thing.
Since the voice is part of one's body, what's happening with the body and mind completely affect your singing. A clarinet player has the option in the middle of the concert to disassemble their instrument, clean it out, change the reed, etc, if it's not working up to snuff. A singer can't do that. Allergies making you croaky? Sing through it. Drink too much last night? Shame on you -- sing through it! Out of shape and can't control your breath? Start working out -- and in the meantime, we'll have to figure out how you can learn to sing through it (without hurting yourself)! Having a bad day? Sorry to hear that - sing through it! Oh, and, ladies, your voice teacher can tell when you're starting your period. Sometimes before you know it.
I often marvel at the conversations that come up during a voice lesson. Many of my students vent about their terrible day -- it's the only way they can get themselves focused to actually sing something. Some of my students dig up some deep, long-forgotten pain that explains why they can't let go and just sing. Others tell me of problems in their relationships, work environments, or other areas of their lives. I have also come to the conclusion that some of the best singers I work with (either in the studio or side-by-side on stage) are the most insecure people I have ever met. In the studio, this means picking up on that insecurity (if it's not overtly obvious) and teaching them to fluff their egos a little before tackling their next song. On stage, it's learning how to give an encouraging word without sounding like you're fake or talking down to your colleague.
With all this said, my students (most of them anyways) are not nut jobs in need of therapy. I actually do have a few that just show up and sing and work on their mechanics with me. The flip side to not putting some of that inner emotion/pain/angst out there is that often times, the students who don't want to share much get the least out of their lesson. Their separation of their emotion from their music can actually hinder them as they are only doing 50% of what goes into making a song good. Which would you rather listen to: a pretty song that's sung with no guts or a song that puts it all out there even if it's not technically spot-on? (If you get both the guts and the perfect technic, then you've got yourself a first class singer).
Part of what I end up teaching is balance. You can't break down into a pull of mush halfway through a song if you're performing it on stage. (In fact, I'd rather not see that during a lesson either). But you need to put some meaning and emotion into what you're singing if it's going to go somewhere. I truly enjoy the challenges my teaching gives me. My own teacher encouraged me to just get out there and start teaching telling me that I would learn more as a performer doing that than she could ever teach me. She was absolutely right. I just wish she told me that the voice-training part was only a small part of of the job.