I'm not comfortable with this as a general strategy stemming from a universal mantra. Learning this as an adult is impressive and probably exceptional, but it's a phenomenon we're all familiar with. We all know people who behave like this reflexively and have done so their entire adult lives. It can be pleasant to hang out with people like that in a heterogeneous group where not everyone is comfortable with everyone else, but they make damned unsatisfying friends and lovers. Policing comfort zones is a sure-fire intimacy killer, and it isn't exactly a triumph of empathy to avoid uncomfortable topics of conversation, even if it requires prodigious skill at empathy. Children of troubled and violent households, such as Bill Clinton, are the poster children for glib empathy that is a mile wide and an inch deep. (No political aspersions meant, by the way; I liked him as a politician and a president.)
I would argue that in this case we are seeing a misguided (and probably temporary) "zeal of the convert", and by convert I mean someone who has discovered the practical value of this Clintonesque "emotional easing." Emotional easing is much like quantitative easing: it shifts unpleasantness out of the current context. It's a valuable tool, but it can't be applied universally. I'm not comfortable calling it empathy, either. "Perking up" when another person's uncomfortable emotions come to the fore and steering the conversation away teaches them that their problems are their own, and that their full selves are excluded while a partial self (with unpleasantness filtered out) is welcome. That's appropriate in some social contexts, but not all, not by a long shot, just like quantitative easing is only appropriate in certain contexts. And it's the opposite of empathy, even though executing it requires a limited use of empathy. Empathy means feeling what another person is feeling because your human bonds transcend the difference between you, not strategically tapping into their problematic feelings so you can override them with your own more attractive and pleasant ones.