Interesting distinction, one that I was unaware of.
I'd disagree with your assertion that lacking cognitive empathy "is clearly a big handicap" and also that "it's definitely not clear whether lacking [emotional empathy] (as sociopaths do) is a handicap".
Speaking from personal experience I frequently fail to exhibit what is deemed to be the correct emotional response to the emotions of others by either misinterpreting their emotion, being unable to adduce the underlying cause for an emotion, or not caring about or being unaffected by their emotion. This has caused much negative interpersonal consequences and has been highly detrimental. I imagine that emotional empathy is quite a big component of what people call emotional intelligence. (I also don't have the little voice that says "don't say what you're about to say" that other people have but I think that's something else?)
However I have no trouble imaging things from the perspective of others. I'd go further than that, I am too good at imagining things from the perspective of others. It's practically a default setting of mine to flip roles and situations around to evaluate scenarios from another's perspective. It's why people with ideological biases infuriate me because it makes them unable to perform this simple "trick", same with someone or groups suffering from delusions of exceptionalism and other tribal biases. Because this is so prevalent and doesn't appear to impact negatively on individuals I have to conclude it's not so much of an immediate handicap, it doesn't appear to hinder people getting on in life the way lacking emotional empathy does. (It does have rather big societal consequences obviously which can eventually hurt the individual via the group but people are too 'stupid' to see this if I may say that without causing offense.)
