Rational compassion: You understand the situation the other person is in and you make socially and emotionally the right decision (based on rational assessment).
Empathy: "the action of understanding, being aware of, being sensitive to, and vicariously experiencing the feelings" emphasis on vicariously experiencing !
Empathy is often used to express that you are feeling the same of similar feeling as somebody else. From the Grammarly post below (from another poster that didn't understand my point):
"By becoming “the wounded person,” he vicariously experiences their suffering. "
Now imagine two doctors, one is empathic to your pain and one understands your pain with rational compassion.
The doctor who "feels" your pain might not be able to operate on you ... as he emphasizes too much. His feelings towards you hinder him.
Not to say empathy cannot be useful and is interesting. Yet, I feel it's overhyped and there are dangers focusing solemnly on it.
As for overrated/overhyped, I think those are dangerous words. Totally dependent on how you rate things, how you perceive others to rate them, and how they actually are. It's almost impossible to have constructive communication about something like that because everybody is operating from different sets of facts.
Your friend gets upset. Why? You understood what they were saying and acknowledged how they were feeling. What more is there to do?
Does that hit home? If not, switch roles, so that your friend is the insensitive person who does understand how you feel, but doesn't feel the same way, because it's not their mother. Being able to be sad for a friend, for their loss, not yours, is a basic human capability, and refusing to exercise it when it is called for is indeed callous and insensitive.