I think that when someone treats “those they love” poorly, perhaps they don’t really love them. They may go through the motions, and say the words at appropriate times, but really they’re focused on themselves and view those other people as resources to use for achieving their own happiness.
If you truly care about someone, his or her contentment and happiness matter to you, and you make those things part of your own priorities. Of course such things aren’t black and white — there’s a spectrum of behavior. Some people may treat their “others” well most of the time, but set their well-being aside when stress levels are high. Some people may just act like jerks most of the time. Either way, it’s a sign that one values one’s own agenda over that of the significant others in one’s life.
There’s a difference between having someone be “important to you” vs. being someone you “care for.” I think many people consider their family members important, but as contributors to their own contentment. Spouses who marry because they want the status, or the financial benefits of “having a spouse.” Parents who seek to “succeed vicariously” through their children’s achievements. It comes in so many different flavors.
All of the above sorts of people I can just dismiss out of hand — one shouldn’t claim “to care” unless one actually does. But I imagine there are also some people who really do care for their close connections, but simply don’t think. Those people are worth trying to salvage.
It’s not that one should value the happiness of chosen others “more than one’s own happiness.” That’s a bad path to be on too — it’s all supposed to flow both ways, and every one of us has a right to prioritize our own happiness. Rather, the happiness of those people should be an important integral part of one’s own. And thought should be given to that fact, regularly. Handle it that way and these things will just tend to go well all on their own. People who pay attention to getting this right, in all directions, become stronger than the sum of the individuals. And that’s the very essence of “family” and “friendship.”