I do tend to give responses like this, but they feel awfully fake to me. I may appear more authentic than I feel when giving them. One time I asked my mother if she would describe me as a warm person (I wouldn't, but I wanted to know what other people thought,) she said that she generally wouldn't, but sometimes I am, and gave an example of a time when she was distressed over a cancer scare, and when she started crying, I immediately walked up and hugged her.

But I also remembered that event very well, and to me, hugging her didn't feel like a natural reaction to consoling someone in distress, it felt like "Crap, I am required to Do Something, what do I do?" and desperately searching for a socially appropriate response.

This probably makes me sound a lot more uncaring than I actually am. It's certainly not that I don't empathize with others' distress, but I'm not nearly as emotive as I am emotional, and I become distressed when I feel like I suddenly have to signal compassion in a way that's different from my response to actually feeling compassionate.