So it seems to me quite likely that modern media consume our "empathy points".

That's a good way to put it.

I've found a Reddit comment that describes another related problem:

This is the internet version of what happened to small towns. People were close and cared for each other because that is all they had. Once you open up the floodgates to a whole world of choice, settling for those around you makes you feel like a sucker, and it ruins you.

Spending time offline is probably part of the solution.