Its rare to find a man who really wants a truly independent woman. That kind of a man has to equally if not more strong in character.
They're becoming a rare breed in today's society.
I think that trait, in itself, isn't all that attractive in a romantic companion. As part of a set of other quality traits, most men think its fine. A woman still has to be pleasant, supportive, fun, etc. Independence isn't a substitute for all that.
Like I said, men and women tend to be rejected for substantive reasons rather than shallow ones. Few women (or men) are truly rejected because they are too successful, have their act together, etc. They just aren't delivering on qualities the opposite sex finds appealing. Independence isn't a substitute for that in any way.