But I think there is a real problem when worrying about one problem comes at the expense of caring about another more severe problem.
Contrived thought experiment: You have the magical power of either curing your own toothache, or curing the world of AIDS. You choose to cure your toothache because it affects you. I would argue you should have considered the alternative.
The problem is not that you can't care about problems that are less severe than others, but given finite availability of "care" in your own mind and in the world, it is better to direct our "care" to where it can have the biggest impact.