I think a lot of things are getting conflated on the "emotional" side.

1 The ability to sense the emotion of others.
2 The ability to feel the emotions of others in yourself.
3 The likelihood of feeling an emotional reaction to the statements of others.
4 The people skills to effectively manipulate the emotions of someone else.

Psychopaths are very good on 1 and 4 but not on 2 and 3.