Empathy is still a bit of a mystery to me. What I do know is that it's a continuum...people have varying degrees of it. It's also not a case of you either have it or you don't....empathy can be learned....the research supports that. Though the person of course would need to have a desire and openness to learning empathy.

I fall higher on the empathy scale. That can be difficult for me but great for others around me. My work has required a great deal of empathy. I have struggled with 'negative empathy' quite a lot. I think that's the main problem for people who consider themselves "empaths" (though I don't label myself that way)....they have lost the line between empathy and negative empathy.
Negative Empathy / Psychology Today
Negative Empathy | Psychology Today

I also think it's important to point out that empathy, or a higher level of empathy, doesn't make someone "better than" others. It doesn't mean that the person is perfect by any means. We all have our flaws. Sometimes I hear people describe themselves as basically saint-like based on "empathy" and that's a red flag. They are very quick to point out the flaws of others and to avoid any self-analysis. I wouldn't call that empathy though it is a story they are telling themselves.

Similarly, someone with less empathy isn't "bad" or wrong. Someone I love very much is quite a bit lower on the empathy scale from me. But he is a wonderful person. And loving. There are just certain things he cannot relate well to if that makes sense. Or he'll need it broken down very explicitly as to how to respond in a given scenario.

Theoretical, you asked about feeling nothing. Great Q. I often feel empty or numb. My whole life. I am told it is because I have the capacity to feel emotions deeply (and h/o trauma) that my mind starts to numb out a while to protect me. I don't know if I believe that but it's what I was told

And yes, we can of course keep empathy in check provided that we want to. Dr. Gabor Mate has much to say on that subject. He noticed in his research and clinical practice that people who put others' needs before their own are much more likely to get serious diseases or to die at a younger age. He wrote an interesting book: 'When the Body Says No.'

The Need For Authenticity - Gabor Mate
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Society tends to skew ideas in very reductive ways. There is actually a middle ground between callous individualism and negative empathy or putting everyone else before yourself. Though for many of us, that's not as easy place to find. And yet, well worth the effort! That's one of my new goals...to put myself first.