In the introduction to his Abelian Varieties book, David Mumford writes:
I don't believe the word "Jacobian" is ever used in this book. Rather stubbornly I wanted to prove that the theory of abelian varieties could be developed without the crutch of "reduction to Jacobians."
Why did he want to avoid this approach? Was it just to demonstrate that you could develop the theory of abelian varieties without using Jacobians? Are there any reasons why you would want to avoid a Jacobian-based argument?