That's not empathy, though. The word "empathy" has been co-opted to mean "understanding someone else's point of view," but that's not what empathy is. Empathy is feeling others' feelings. I'm actually empathetic in that I sometimes experience an emotional response (limbic) similar to an intense emotional response I witness others having, especially if they're a person close to me. This is very different than making a conscious attempt (prefrontal cortex) at intellectual understanding of someone's emotions.
Sounds like cognitive empathy vs affective empathy.
I'm not so sure I agree - well maybe I do, I meant literally feeling in my statement not merely understanding. e.g. I eat meat - but I can literally feel the cringing sadness and disgust of a vegan if I imagine their perspective, even if I disagree.
I don't think it's been co-opted? Mirroring the emotions of another person you're actively observing doesn't give you insight into why they're feeling that way. It's just mirroring, but its an excellent starting point for learning. To have empathy, for people you're not actively observing, or for future states of people you are observing, you have to be able to model them first, and then mirror the emotions that the model predicts, which can then update the model. This loop is empathy, its both "experiencing other's emotions" and "the ability to understand and predict".