I agree with Salmon that not chatting about weather or whatever does not mean impersonal or cold. My guess is that the T you're talking about is simply neural at this point, which makes sense as you don't know each other and haven't established a relationship yet. The genuine warmth requires that the relationship has developed to a certain level, otherwise it's just a pose.
It seems that you value highly this pose but somehow confuse it with caring and warmth.

To answer your question though then yes, probably most psychodynamic therapists would not do those superficial gestures as this is really not the point of therapy, nor does it necessarily help you in any way.