According to a previously asked question on MSE (Is $M_g$ NEVER proper? And why does $T_g$ contain products?), the Jacobian of a curve cannot be a product of principally polarized abelian varieties. What is the reasoning behind this statement?
I know that the closure of the image of the Torelli map $\mathcal{M}_g \xrightarrow{j} \mathcal{A}_g$ sending a curve to its Jacobian is made up of products of Jacobians, but it's not clear to me why the actual image of the map can't contain such points (or of principally polarized abelian varieties in general). I did find a paper which had a generalization of the Torelli theorem (i.e. $j$ is injective) for a map from a compactification $\overline{\mathcal{M}}_g$ which seems to indicate that two stable curves can have the same image in this extended Torelli map only if they are both singular if at least one of them is singular (actually have more precise conditions -- in fact, the authors obtain an if and only if condition: see http://www.mat.uniroma3.it/users/caporaso/tostJEMSproofs.pdf for more details).
Even with this, I guess it doesn't really finish the question since there are a lot of principally polarized abelian varieties that aren't isomorphic (or even isogenous) to a Jacobian.
Is there a simpler reason why the statement should be true? If so, how should I think about this? Also, does paper mentioned above actually apply to the situation that I'm thinking about? I'm a little worried since I may be mixing up different types of compactifications of $\mathcal{M}_g$ here.
Update: It should be made clear that we are probably taking the product polarization of the product of principally polarized abelian varieties mentioned in the old MSE question. Otherwise, it may be possible to get a Jacobian which is isomorphic to a product of Jacobians. In the case where $g = 2$, we don't even need to ignore polarizations.
In fact, products of elliptic curves $E_1 \times E_2$ with the product polarization are the only products of elliptic curves treated as principally polarized abelian varieties which are not isomorphic to the Jacobian of a genus $2$ curve with the associated canonical principal polarization (see ``Zum Beweis des Torellischen Satzes'' by Weil). This can be confusing since the only polarization each $E_i$ has is principal. This still doesn't explain why arbitrary products of principally polarized abelian varieties with the product polarization would be bad.