I have noticed that people who claim to feel empathy with people they don't know and try to do good in far-away places and otherwise seem to want to shoulder the burdens of the world don't make very good friends, as if they are incapable of caring up close and personal. Maybe they are the ones suffering from Empathy Deficit Disorder.
Having said that, I do think something like empathy exists, but is limited to the skill you develop in guessing how a close friend feels, assuming we all feel roughly in the same way, to be able to help the ones you care about better. Note that I care about very few people.
So, no magic resonance or ennobling trait, just a mechanism that helps you be a friend and have friends. I also fail to see how this generalises to the rest of the world. You would probably say this "limited empathy" also stems from selfishness, but I think we agreed to disagree on that before.
There's still the question of the negative: the total lack of empathy for anyone, which would make you a sociopath, for want of a better word. What does somebody who isn't a sociopath have, if not empathy? Maybe we just need another term for this.
You can care for someone and still not feel empathy. Caring is more or less self-interest for your own social experience. It has nothing to so with empathy.
I don't think sociopathy exist. If it does then yes, we are all sociopaths. The question is always "in regards to whom".
Trying, however incompletely and clumsily, to mirror in your head how somebody you know might feel, by observing well, could make you better at finding out what is wrong, enabling you to care better. But I will admit that is perhaps not the usual definition of "empathy".
Dogs do the exact same thing. Babies also do it. If they see their mom in distress they also cry but it doesn't mean they have empathy. We trained dogs to be perfect mirrors of us. This is why they seem so "human" sometimes