Perhaps the sincere words, "I'm sorry" open the door to the action and the healing, however minor or major.  If someone is going to say the words but not mean it, I don't want them to say it.  But if they are sorry, it helps me to start the process of getting beyond it even though I will get there eventually either way.  For myself, I can be sorry, but can't overcome the other emotions right away to express it.  I do try to tell my kids I'm sorry or I'm wrong when I need to.  I think it's healing and a bridge.