I think that you have a point, that she is possibly overwhelmed in her own life/schedule.
Much of what you relate, to me anyways, screams "I can't humanly take care of you to the extent that you need it". She probably cares very much but sounds stressed and yes, defensive. I have found it helpful to point out defensiveness to T. Usually, T will go back and think about it.

Sorry, this sounds painful. I've had times where T has told me to go call friends--that hurt!! The things I needed to talk about were not friend-safe topics.