They have a conscience, they have empathy, it's just that they only do for their friends, families or people in their social circles.
Most people act like this. There are people working in terrible conditions in Bangladesh or China to create stuff the western countries buy at Walmart, forever 21, Amazon... Where's the empathy for them? Where's the empathy for the guy that lost his job 6 months ago and has been sleeping in a tent since, and only got 75 cents after 2 hours of panhandling?
Individually, we can't help everyone so we often look the other way. That doesn't make us psychopaths.
Furthermore, saying rich people have mental illness frames the problem as a psychological one when in reality it is a socio-economic problem that requires a political solution.