I wonder if those that are sensitive, those who experience emotions so deeply, experience life to a greater degree. In feeling more, they experience more. This heightened experience feels precious.
Some of our best memories in life are definitely tinged with emotion. I would rather be prone to deep emotion - even if it makes for some awkwardness. Better that than the alternative, like colour-blindness for the heart.
PS Hiding your emotions (by acting otherwise) is another thing entirely. Here, I talk about what you experience internally, not what the world sees of you afterwards.