I think its rude to casually ask people how they're doing.

It seems to be considered common courtesy but I don't think it should be. Its rude to casually remind people of all their problems if you don't want to stick around to hear about them.

Its different if they actually care about your response, but most of the time you have to lie through your teeth to give a expected response of "good, how are you?" And most of the time they don't even answer because they're already walking away.