I did briefly try out some therapists who were kind of like what you describe. They seemed to be just playing a role, or going by some technique or manual they learned in school somewhere about how to look genuine or empathetic or whatever. I didn't stay with them beyond a few sessions. I suspect their style would have worked for someone out there, but there just wasn't any kind of connection that gave me any incentive to return.

The three therapists I worked with long-term never gave me that feel. They were very natural in their interactions with me and their level of care and concern was easy to see as quite real and not at all staged. They cared beyond our session times, and there was plenty of evidence for that. It never crossed my mind that they were just caring as part of their job. We connected very naturally from day one and worked well together.

We won't mesh with everyone with work with. Perhaps it isn't so much about them not caring or them putting on an act as much as it is simply that something about personalities just doesn't gel with certain people. Personally, I would have trouble working in a therapy relationship with a person that I just felt rather ambivalent about, so personally, I have always felt comfortable continuing to look for a therapist that clicked with me better. I generally didn't have to look that long to find them. In fact, the three the really did click with I found practically without trying at all weirdly enough.