How can you possibly create a controlled enough experiment to give quantitive results? You’d need two different leaders leading the same people and companies to be able to compare.
I may not have an absolute answer but I think it’s not impossible to measure the leadership styles. There is a lot of survivorship bias in leadership stories and acknowledging that is a very first step to measure it. To me a good leadership style inspires and help others to become a leader particularly those who at first glance don’t seem like a leadership material. We can simply count the number of people one personally guided to either become a leader or grow as leader. We can also measure the impact they created in their own functional area as well as in whole business unit.