I too think some do and some don't. I also think that each therapist both does and doesn't care about certain things. And that they have personal reactions to triggers that they are not always aware of, and that these can sometimes lead to less than ideal responses to clients. And I think that one client perceives as 'caring' might seem completely inadequate to another client, and completely invasive / inappropriate to another. I think my long term t cared deeply about me, but there are circumstances when her caring was not apparent at all. I think Ts can care deeply about a client or situation, and yet still respond to something in a way that suggests uncaring instead.
I think there is no one standard of what 'caring' looks like, and not one T in the world who 'cares' in 100% of the ways we would wish them to, or is able to sustain caring behaviors 100% of the time.
When it comes to therapists I don't think we can speak of them as a 'they'.