I disagree, too.
I am also a teacher and deeply, deeply care for every one of my students. I feel a real emotional connection to them, and their well-being - physically, emotionally, intellectually - is so important to me. When one of the children in my class is hurt, sad, angry, struggling - whatever - I feel for them and want to do everything in my power to help them deal with whatever they need to deal with. Yes, the caring is shared. But when I spend time with each child, listening to them, hearing them, sharing wtih them, I am totally there in that moment for that child. (And it is real!!)

I can't speak for other people but I know for myself, when I get to know someone I care about them. I can't imagine getting to know someone emotionally, as is done in therapy, and NOT caring about them. No, therapists don't care about clients in the way that a person cares for their spouse, or sister, or friend, or child... but I am sure that most certainly do feel genuine care and concern for their clients.

Xtree, you wrote: Ts cannot possibly care for all their clients anymore than my doctor or my lawyer.

I disagree, and here's why:

Doctor's care about the health of their patients, and help by handing out advice and medicine.
Lawyers care about the legal needs of their clients and help by advising and assisting clients through the legal process.
Therapists care about the emotional lives of their clients and help by facilitating change through a reciprocal relationship and emotional connection. That in itself involves a genuine feeling of care for the person they are helping.