Not sure what the downvotes are for on this one. It depends a lot on what "genuine care" is supposed to mean. If you want to interpret that as a subconscious feeling then you're right. Feelings aren't normally controllable and calling them up on demand is pretty much impossible.

That being said, if you go through a bit of game theory and apply it to the real world - the experience of the last few millennia of recorded history is the strategy most likely to get people what they want is lots of communication and setting up win-win deals for everyone. Someone who reliably offers win-win deals has a natural advantage over the more common person who thinks in terms of win-lose deals. Communities that make a habit of setting up win-win deals for their members have an overwhelming advantage over those that don't. If you tap in to that type of thinking it tends to translate into taking a real interest in how other people are going because it is easier to set win-win deals up if you know what their problems and goals are. And a sensible sub-strategy is making sure to be as kind as possible to everyone to get into the habit of thinking empathically and keep channels of communication as open as possible.

So if "genuine care" means you literally feel something... nobody has much use for your feelings, we can't tell what your feelings are anyway and you probably can't call them up on demand. If "genuine care" means you try to figure out what other people want and then help them get it then that's simply good strategy and most people should find their way to it if they think about it for long enough. Some people have to think a bit harder than others and there are a few rare maniacs who really just want to cause pain and suffering. The maniacs are bad news.