Great post. I just want to leave a #comment
As you have explained both etymologically and in terms, both sympathy and empathy play an important role in socialization. Socializing with friends or even relationships between couples still require both.

Maybe we often have difficulty distinguishing between sympathy and empathy which actually have differences as you have described. Sympathy grows in a person, maybe because they have been in the same position as someone else. So he will show concern by giving words of concern without being followed by emotions by focusing more on giving superficially because it is more on the surface.
This can be shown by several attitudes such as giving judgment and then giving advice when chatting, having thoughts about what others feel.
Empathy gives a deeper impression because it involves emotions so that we can feel what others feel. It can be said that empathy really involves our emotions towards others. With this emotional involvement, attitudes will emerge such as being able to find other people's perspectives, being able to listen better to what others say, being more of a non-judgmental listener.

And I think what you described is right. Empathy and sympathy play an important role in our lives in this world, because we are social creatures who cannot be separated from being together with other people.
Warm regard from Indonesia
@rokhani