Because if you care about someone else (i.e. put a value on protecting and aiding that person), you become a resource worth preserving to that person.
People don't know how to pretend to care, thus them being terrible at it - see, for example, not even spending five minutes to try to think of a way to bring their friends back to life.
It's worth noting that in the Milgram experiment, there is no perceived punishment for failure to participate, just a polite repetition. Further, the Milgram experiment models willingness to stop acting in accordance with orders, rather that willingness to act against orders, which, while morally fairly indistinguishable, are psychologically (and legally) substantially different.
It's my understanding that people prefer to go counterclockwise in all sorts of situations, it might just be that.
Baba Yaga has "been dead for six hundred years," and a quick Wikipedia search suggests the historical myth is first recorded in 1755, nor can I find anything particularly relating her to being from around ~1400. Nicholas Flamel is six centuries old (canonically, he was born in 1327), which means the Philosopher's Stone, if it exists, is around the same age.
Not sure what kind of coherent theory you can come up with to put it all together, though... Voldemort = Baba Yaga seems a little... silly, especially given Quirrell talking about female wizard rapists, which, given that Canon!Voldemort is a rape baby and Quirrell is Voldemort, seems like pretty good evidence that HPMOR!Voldemort is a rape baby too.
Maybe Baba Yaga is Nicholas Flamel's true identity.
From chapter 85:
"And the last was cousin to your young friend Lavender Brown, and he -" The old wizard's voice cracked. "He did not return, did poor John, and he saved none of those he meant to save."
Did anyone else get this ref? I haven't seen anyone else post about it.
I would say the likeliest explanation is that people do care, but only insofar as it enables them to signal that they care. Caring much farther than that is pretty much pointless, from an evolutionary perspective, and probably actively detrimental.