I struggle with cognitive empathy but I effortlessly have emotional empathy. In fact, I have to 'manage'/reduce my empathy levels so my emotions don't cripple me.
Good article on this topic in regards to narcissists - they have high levels of cognitive empathy & know how you are feeling but cannot 'feel' your pain or emotions.
They recognise your pain but they don't feel your pain. If they did, they'd want to help you.
https://narcissisms.com/do-narcissists-have-empathy/