Yes!

I don't think they're paid not to care. My T cares about me in a very genuine way. It's not my job to take care of him though and so yes, his stuff is kept out of the room although it comes in at relevant times and because I'm nosy

Either way, I think what makes an ideal therapist are the characteristics above, plus training and accountability. My T has all of those. He's not perfect but six years in, he's demonstrated an competency in his field that leads me to trust him.