There perhaps is not a definite answer to that question, even with generations of study donated to leadership and with lots of people approaching the question. Why? Arguably, leaders aren't 100% made or born either way. They're a bit of each.
If that's the case, then why do people even ask the question? Why don't people ask "are videographers/solicitors/vets born or are they made?" as much as they ask the question about leaders in particular? It could be argued that this is due to the fact that leadership is viewed not only to be difficult, but that it's very hard to do to do properly. Therefore, people expect that only a special type of individual can successfully assume a leadership position - whether in politics, business or something else - and that the person must be born to assume the role more than created and crafted for it.
Leaders: Are they born?
It can be argued that leaders are born in the way that they are born with the ideal qualities and characteristics. For instance, it is obvious to think that an introvert would be less likely to assume a leadership position than an extrovert, primarily due to the fact that they might be less confident and less personable, hence looking like the extrovert instead was "born" for the position while they're not. Of course, there's nothing saying the introvert will certainly not be an effective leader or that they will not do any better in leadership compared to their extroverted chum - they may even excel them.
This might be where the logic behind "natural born leaders" comes from - traits such as intelligence, charisma and confidence can definitely help someone to become a leader, but they're not the leadership be-all and end-all. Not only that but there is nothing to say that individuals lacking these types of qualities early on in life will not be able to improve and develop them at a later stage in their lives.
A leader is made...
The argument that leaders are made (not born) is much more compelling. We all need to work on and improve our skills, whatever it is that we do - a "born" leader thrown into their first ever leadership role might fall at the first hurdle due to the fact that they don't hold any hands-on or real experience. Meanwhile, an individual is far more likely to be "made" a leader, as they will be crafted into being a leader with the right training and development and with the correct kinds of people leading them.
However, not everyone is able to be made into a leader. What is that saying? "You can lead a horse to water..." If somebody is unwilling to devote their time and themselves to leadership development then it is unlikely that they will ever become an effective leader, no matter how much money is spent on leadership training for them. Particular qualities - perhaps those that they are born with - will certainly help to influence whether someone's able and willing to do the position, such as intelligence and charisma. After all, the person may not want to do it, period. If that is the case, what can you do?