Having a social relationship with someone != caring about their fate.
I think you aren't understanding the mechanism of the theory - which goes beyond the original paper. It has less to do with being able to "be social" (a broader function) than it does with being able to empathize with someone. Empathy is directly related to size and activity within the neocortex.
Dunbar's Number is "a suggested cognitive limit to the number of people with whom one can maintain stable social relationships."
Having a social relationship with someone != caring about their fate.