To an extent, I partially blame Jennifer Lee for deliberately not fleshing out Elsa further. Lee, having been a younger sister in real life, related to Anna more and actively pushed for her to get more focus in the movie. It wasn't until the Lopezes wrote "Let It Go" that they realized they had something incredibly unique and special with Elsa.
I actually love Anna and think she is unfairly, though understandably, overlooked. But Elsa is a type of character we just don't get in Western animation. In many ways, Elsa is the inverse of Riley. The nature of her powers are such that her emotions are externalized and actually have the capability of affecting people on a large scale. That amount of simultaneous vulnerability and power has immense potential to be explored further. She runs because it's not just her life that is at stake but also that of her sister's and the entire kingdom's. Her demons have been laid bare and they are triggering an icy end to the world as she knows it.
Here's hoping we get to learn more about Elsa's incredibly fascinating psyche and powers in Frozen 2.