I feel that a woman who pursues pure "independence" is a hassle to deal with. I think that there are rare cases in which a "successful" woman will still [consciously] want the leadership of a man.
My uncle and his wife are one of those cases. Granted...they're both Chinese and completely not Westernized. She owns her own property investment and management company and my Uncle owned an even bigger one. I guess they partnered up and she basically became his little secretary for life. They look like the typical traditional Chinese family in which he calls all the shots while she sits at the desk doing paperwork for him.
Looking at them upfront, they seem to be a smooth-sailing couple. If they ever do have discrepancies with each other, they do a good job at hiding it.
Edit: As for the original question. For me, if a chick has that aura of not needing a "man" and is an "independent, successful" woman, then I'll just stay away. I know one woman who's like that, but I want nothing to do with her romantically (can't say the same thing about her to me, though). The important thing to me is that a woman wants leadership from a man. In my experience, her conscious desire for a man decreases as she becomes more successful and independent. As she becomes older and unable to secure a man, she'll become lonely and the dissonance between her thoughts [rationalizations] and biologically imperative emotions will start to surface.
I don't mind a successful woman, if she comes from the very small percentage of them that knows the true value of a man and understands that she needs one.