If leadership means having an ability and desire to inspire and influence others then I can see the argument that leaders are born with leading qualities. For example when I see my sister using her inherent ability to influence our younger brother (and sometimes our parents) to do the things she wants them to do.
If leadership means being courageous and willing to speak out for the betterment of those around you, I would think that leaders do not have to be born as such. I have seen people who consider themselves introverts and overall followers become successful leaders when faced with an issue they are passionate about. If we define leadership as having a certain intelligence, creativity and/or drive, then I am back to believing that it’s a trait you’re born with, but you may or may not continue to develop that trait through your life.
We all have areas of our lives where we have talent and tendency for success. If this is also an area you feel passionate about, you may exude qualities that are absent from other areas of your life. So while you may not be a natural born leader you can certainly overcome many obstacles and develop a desire and ability to lead when you are inspired to do so.