Earlier quoted context omitted.
My read on the book was "humans are really good at telling if you genuinely care about them or not and will respond well to that, so you should genuinely care about the people around you, and good things will result from that overall, especially if you're not super mercenary about it." Bill & Ted said it most pithily: be excellent to each other.
Serious question - what is your definition of "genuinely care"? The Carnegie example doesn't show "geninuely care" to me. It's nice, I think I should do it. Give people random but geninue complements. It's nice. It costs nothing. It makes both of us feel good. But, is that "genuinely care"? I ask because I'm bad at conversation. I hear this "genuinely care" and I just, usually, can't get myself to do it. I don't care…
I think your comment reflects that you're waiting for someone to say or do something which will cause you to care. And that's effectively waiting to get something from them. You need to cultivate the sense that everyone in some sense has the same daily struggle, and be the bigger person who strives to alleviate some of that loneliness and suffering in others.