Sympathy means expressing pity or sorrow at someone's plight. Reflecting feelings is more like a form of empathy. It's clarifying and/or paraphrasing the feelings so that the other person feels like they're heard and taken seriously. They're orthogonal behaviours - you can do one or the other or both.
I agree that I should understand the other person and take them seriously, and to convey this to the other person. However, sometimes verbal reflection serves the purpose of blind affirmation. I don't think the verbal component, construed this way, is so productive for empathy.