You can genuinely care in one moment and forget about them five minutes later, that's ok. Part of making conversation is also stopping the conversation when you are done, not letting it bleed out. My favorite is: "I really enjoyed this conversation, now I'm going to read my book".

I find it nice to connect to strangers in real life, if only for a moment. It can be about something silly as sharing a very bright bird you see while waiting for the bus. It can be giving two dollars to the woman in front of you at the grocery store, cause she's short.

Also, having this connection with people when it is about nothing (small talk) helps build communication skills you need when it is about something.

I genuinely hope you will get answers on your question, maybe even in this thread. But I'll also forget about it in two minutes.