Both, of course.
You can teach them to be brave. You develop their definition of ‘normal’, develop their trust in you and teach them to grow their confidence in you and themselves. You take them places, show them the trash cans on garbage day and teach them how to be responsible, respectful equine citizens. This is the foundation of bravery, laid over the course of time with 100s of tiny decisions being done daily.
More important than bravery, in my eyes, is their sense of self and security. Are they OK being in their skin? Do they think they are hot stuff? Do they have the innate confidence - that look of eagles - that you can harness and develop? That self confidence allows you to mould them into a secure horse, one that doesn’t require all their friends and familiar surroundings for them to be ok. In my experience, some registries tend to value that more - i.e. Holsteiners in my experience have a very level head on their shoulders; thoroughbreds typically can be more reactive - but - with proper training (i.e. the ones off the track), they can be very workman like and understand what is required of them. That can translate into a perception of brave as they know what is expected and trust they can deliver it.
My best have been those that were born naturally curious. I worked carefully to nurture that, not quash it. Give them the time to investigate, reward them for wanting to go see that strange object in the corner, careful to not inadvertently reward the spook (critical difference). But I will tell you, as youngsters they were sensitive, hot and reactive. I really believe that our bond, knowing that I have their back and will never fail them, is what allowed them to develop into the confident horses they are. Had they been pushed too fast, over faced or inappropriately challenged they would not be the steady eddy types they are now.
Interestingly all of mine have very similar temperaments, despite starting out innately a bit differently. They’ve learned how we gel and have become a part of the fam; it’s how we roll. LOL