This probably makes me sound a lot more uncaring than I actually am.

Not at all. It makes you sound exactly like I feel a lot of the time–as someone who didn't naturally pick up a lot of social scripts, it just feels frustrating that people have these scripts, and expect you to know when and how follow them even though they're completely counterintuitive, and that people care about how you appear, not your intentions (or what you actually get accomplished).

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.

Fake it till you make it! And take this as consolation: plenty of people's natural, instinctive responses to people in distress aren't helpful. The fact that you're actually thinking consciously about your response means you can notice over time what works and what doesn't and adjust accordingly.