I would tweak your conclusion slightly. In my opinion, the simplest way to understand it is that people who are emotional want their feelings validated. It might seem like I'm splitting hairs, but I think there is a slight improvement in clarity to say that what you're describing is the process, the goal is validation.

The other person wants to know that their communication has been received, understood, and the emotional content validated. I believe the goal was achieved in your story by your focus on solving the problem -- thereby validating the emotion of the other party because your response indicated there was in fact a problem thus the concern was valid.

The process will never be as effective if the receiving party only reflects that they understand the emotion exists. "I hear you're concerned," by itself has some value, but far less than, "i understand your concern. There's a problem here. Let's work together to resolve it so that you needn't be concern anymore." The latter is far more effective because it acknowledges the essentially rightness of the feeling.

In his seminal book "How to Win Friends and Influence People", Dale Carnegie has a novel approach to the problem. He recommends that in every case we can sincerely say, "If I were in your position, I'd feel exactly the same." He rather cunningly points to be in the other person's position, we'd have to BE the other person; thus we would experience precisely the same feelings.