I agree with those who suggested associating calling a no-show client with caring is not so much more than subjective projection. I personally feel that chasing a client who decided not to go to session is intrusive and it is more respectful not to interfere with that choice, especially if the client pays for the session anyway. I think when Ts call, it's probably more often for them, they want to know whether the client will still come later or needs to reschedule so that the T can adjust their own work schedule. Maybe they do worry sometimes, for example if the client exhibited suicidal thoughts or other harmful behaviors recently, but I doubt this would be the very first thought of a T in general, without reason.

I never no showed in therapy without notifying the T but one of mine was quite bad at administrative things and somehow did not register my advance cancellations three times. He would then call me at the time when the appointment was supposed to happen and it always made me a bit angry. I also did not like when he would call me on the phone out of the blue because he was upset or made completely false assumptions about me.

For me, if anything else, a sign of caring is when someone seems engaged and interested in our interactions when they happen but respect my choices when I don't want to do it and won't intrude. It is indeed interesting to think where these subjective interpretations come from. For me, very definitely at least in part having an overly anxious, often intrusive mother, who did not easily understand/accept my wanting privacy when I needed it and the only way to deal with her was boldly rejecting and avoiding her. If someone often had the opposite experience, not getting enough attention when wanted, it's no surprise that they associate it with lack of caring. But in reality, it can be caused by a lot more different things, including actually respecting the other person's privacy.