My two cents, as someone who feels empathy and doesn't like it. I'm answering to different replies I read without quoting the authors, in order to do a general reply.
Empathy exists, and it feels like an emotion, not like a thought, even if it's different from your own emotions. It's automatic and deeply intertwined with guilt and with shame. Sometimes it can be suppressed but it can't be done voluntarily (you need to be angry or dehumanize. Maybe also a big enough fear works).
People with empathy don't care more than others about strangers until they interact with them. If I am walking and I see a beggar I don't empatize with him, but if he talks I do. Empathy doesn't always work in a prosocial way: sometimes when a beggar talks to me I give him money, while sometimes I feel the need to distance myself from that source of unhappiness so I can react by walking faster bowing down my head a little because I feel ashamed of the disgust/anger that feeling his unhappiness gives me. (Ugh, why does he give me his unhappiness and tries to make me feel embaressed/guilty?! He is doing that to get money... ###$ him! Ugh, I'm a piece of $#%^ for everybody to see... ###$ him!)
People can also care about strangers without feeling empathy both for ideological/identitary and for narcissistic reasons. Usually it's out of repeated experiences of empathy that people develop this kind of ideology, and then they apply the ideology even when not supported by their emotions.
Empathy is a survival strategy. Feeling what others feel can save your life if you are a child in an abusive environment: you learn how to please others, how to diffuse or avoid conflict, how to anticipate what's coming. You are focused on how others feel because you know that human beings are dangerous. You learn to like pleasing others (since you have to, you may as well learn to take pleasure in that) by feeling their pleasure. You learn not to anger others because you can feel their pain.
Narcissism... Is similar. It's shame, shame, shame. Shame is another negative emotion that protects you by keeping you likable and in line.
I say that empathy is a survival strategy because it seems correlated with internalizing disorders, and it has a negative correlation with happiness. It makes sense to me: when you feel happy and powerful, other people are not that dangerous anymore (and it's difficult to empatize with pain if you don't usually feel it). It makes sense also because looking in the minds and hearts of others is the basis of self doubt and low self esteem, of guilt, and, as most here on the NPD forum know, of shame.
They say that we are not who we think we are or who others think we are, but we are who we think others think we are. This is the basis of shame, and you can only look in the minds and hearts of others if you have empathy. You can do that with logic instead of course, but logic is concerned with causes and effects and solutions, not with feelings. If you look in the hearts of others with logic, you are looking "from outside", and you won't feel shame. Logic asks "what does this person want?" While empathy asks "what does this person feel?"
Empathy can have a huge influence on behaviour. I don't care about the millions dying somewhere in the world right now, but I can't bring myself to damage someone intentionally in person. I know I couldn't kick a cat even if for some reason I wanted to, unless there was a very good reason to (like my life depended on it). There is a wall that tells me "That's not fair. This cat trusts you. It would suffer", but it's an emotional wall, not a cognitive choice. In fact, I eat meat without blinking. Of course it makes no sense. It's like just interacting with another being creates a little bond in seconds: it's like the cat is part of my in group, while the cow I eat is from another tribe. Empathy and the ability to connect emotionally are intertwined. Empathy means being a little enmeshed, not separated. It creates interpersonality, a "space" where two people are tuned together. Empathy is the opposite of being... Strangers.
Shame is a huge motivator for me, but even if no one is looking, if I try to break this wall of empathy and guilt (when I feel it, irrationally) that keeps me from doing some things, it's like the wall grows stronger the more I resist it, like trying to reach the speed of light... You can go infinitely faster and using infinite more power but the speed limit of the universe stays out of reach. It's literally fighting against yourself, you can't win because the opponent is exactly as strong as you.