How does Bilbo's dramatic spider rescue show how far he has developed?

The spider incident finds Bilbo the only one of the group capable of saving the whole. He has conquered his fears and responds proactively, meaning he has a purpose for his actions...... they aren't merely reactions. He also begins to think rationally and act methodically. Let's face it, when Bilbo was attacked by the spider, he reacted from fear. Killing the spider was a defensive move, after which, he passes out. Freeing the dwarves from the web was a planned maneuver. Bilbo is using his brains.