There are two kinds of, "I'm sorry."
The first kind is the apology of responsibility, of blame and of litigation. It is the four-year old saying to his brother, "I'm sorry I hit you in the face." And it is the apology of the surgeon who forgot to insert sterile dressings and almost killed you.
The other kind of sorry is an expression of humanity. It says, "I see you and I see your pain." This is the sorry we utter at a funeral, or when we hear that someone has stumbled.
You don't have to be in charge to say you're sorry. You don't even have to be responsible. All you need to do is care.
In this case, "I'm sorry," is precisely the opposite of, "I'm sorry you feel that way," which of course pushes the other person away, often forever.
As we've been busy commercializing, industrializing and lawyering the world, countless bureaucrats have forgotten what it means to be human, and have forgotten how much it means to us to hear someone say it, and mean it. "I'm sorry you missed your flight, and I can only imagine how screwed up the rest of your trip is going to be because of it."
"I see you," is what we crave.