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?
The Jacobian of a smooth projective curve $X$ of genus $g$ is indeed an abelian variety of dimension $g$, whose points parametrize line bundles of degree zero on $X$.
But there it is quite false that every abelian variety is a Jacobian: recognizing Jacobians among abelian varieties is the Schottky problem (actually it is a bit more technical: you have to take principal polarizations into account).
So Jacobians do not suffice for studying all abelian varieties.
Moreover defining Jacobians purely algebraically is a very difficult problem, first solved by Weil.
He invented abstract algebraic varieties, defined by gluing, precisely for the construction of the Jacobian: before him varieties were supposed to be projective, i.e. embedded in a projective space .
(Actually the Jacobians are projective after all, but that was proved only later by Chow)
So, no, it is not satisfactory to study abelian varieties through Jacobians and Mumford chose the right approach... [surprise, surprise :-)]