I love talking about the emotional component of singing. Recently, a voice coach said 'in 98% of these cases, the only reason the singer can't do what they want with their voices is because of ineffective and inefficient voice teaching..'
I agree that many people have been given incorrect singing habits thanks to a teacher, and it's a process to reprogram positive vocal habits, but what I have found is the converse of your comment. I believe that most vocal problems have a root in false beliefs or emotional blockages, and that no amount of technique can overcome a subconscious need to hold, constrict and protect - which manifests in vocal challenges.
Usually, voice work alone is the mechanism that finally brings awareness and opens the channels of release - it is that powerful. When the intention is to open and release the voice (which is always the intention!) the very act of getting out of the way and letting the voice 'go' can cause huge emotional release. Often my clients burst out in tears at that point of release, and after that, the technique can then be applied without the past resistance.
I agree, that the intention is never to psychoanalyze the client, and one must be very careful to let the natural course of 'letting go' lead any discussion that may be initiated by the student. Then, I find that being a good listener; suggesting to the client how the emotional issue has manifested in constriction etc.; and then wholeheartedly using singing as the tool for further expansion of voice, and self works very well. When clients get to that point I find that they are thrilled to 'find their voice' and want to open up further.