You might say that C-3PO never completely learns courage. It's true that he complains, but due to various external and internal pressures, he usually ends up doing the brave thing anyway. However, he seems to always be perpetually "mid-way" in his Character Arc: first complaining, then doing the brave thing, but never totally becoming the unflappable hero. Although I can't read George Lucas' mind, I suspect Lucas didn't want to lose the comedic benefits that came from C-3PO's continual anxiety. I think most people would agree that it was a good choice. Also, Luke, Leia, and Han were all heroes of different kinds; C-3PO's worry-wart style of heroism rounded out the mix.
Giving a character we care about an FLBW, and then watching him or her grow through a difficult and stressful Character Arc, is one of the key ways of making a story emotionally gripping.
If your game has major, recurring NPCs, then, ideally, at least one of them should have a Character Arc. It's fine if others do as well. If you don't do this, you might be bypassing a major opportunity to help create an emotionally gripping game.
Again, this is a general guideline, not a rule. There are always exceptions.