I love this question! A great spin on history... thanks!
Here is my answer to your provocative insight:
Obviously, everything about our past helps to weave the tapestry of our present, so I don't think we could EVER be who we are (and likewise, the three figures you mentioned could not have been who they were) without every inch of the past. But more interestingly, I think your question is also asking WHAT ABOUT the depression helped to forge them in a manner that eventually became the great men we recognize?
I think that for Byron, depression acted as his muse. Heightened emotionality tends to afflict every great artist, be they in the visual medium, written, or musical. Byron was drawn to several (many???) romantic relationships - one of which may even have been homosexual - which both fanned the flames of his angst, but were probably also the result of it. In the end, this angst became the gasoline for his creative engine. There have been several research studies which have linked creativity to (primarily) bi-polarity!
As for Lincoln and Churchill, nobody can deny the parental modeling and genetic robustness that allowed for them to grow their leadership abilities. But depression and melancholy can also fuel an increased empathy in the afflicted. In fact, Lincoln was so tall and ugly (sorry... but you know what I mean!) he probably felt constantly like an outsider and lived with a certain level of emotional pain while growing up. Again, this builds empathy, and sets the stage for leadership because these men knew how to connect with their people, knew how to "read" their people, and knew how to act in a manner that would be maximally effective to initiate change in their people.
There are probably another ten or fifteen ways to answer that question because it is one that gets the creative juices flowing! But here was my immediate reaction... thanks again for asking! If you are satisfied with the response, please hit "Accept." That is the only way I can receive credit for my answer. Thanks-