Do you think you would still be submissive if you were not broken? If a magical cure for all your woes were available and you took it, would you then leave your Master/Mistress and lead a 'normal' life as a vanilla? I think this question is the crux of the matter - direct cause/effect relationship between psychological state and submission or happenstance?
I do not have a Master, so my answers may be odd.

Find me a person over the age of 16 who does not have some some sort of emotional baggage, and I'll... do something. I don't think the broken/damaged feeling has an intrinsic relationship with being submissive, but it seems to be common factor. I can't remember ever not feeling some sense of brokenness inside me, so I can't say if I would be the same person at all, let alone if that one aspect would change.

I do not have a desire to be "normal." Ugh. Maybe it's just the sick emotional masochist inside me, but I don't mind calling myself broken or damaged. Now, hypothetically, if I did get "cured," I doubt I would leave my hypothetical Master because I would have chosen him for reasons other than just because he made me feel better.

I don't think the damaged/broken feeling is a cause or effect of submissiveness. There are plenty of broken people in the world who aren't submissive, and from what I've seen here, there are plenty of submissives who don't identify as broken. That said, it does seem to be a common feeling, if only amongst those on this thread.

It's funny - most people probably wouldn't pin me as being submissive. Or broken. Nothing in particular has ever happened to break or damage me - just little things in my mind that won't go away.