What does the Tannakian formalism reconstruct when fed the category of chain complexes?

Deligne does not do what you seem to want, which is give a theory internal to super vector spaces. To do so is probably an open question. The obvious version of Hopf algebras in SVect does not work: the category of comodules admits an action by the lines in SVect (commuting with the fiber functor), and thus cannot recover categories like ordinary vector spaces.

Deligne's theory is external, mixing Vect and SVect. A supergroup is an algebraic group internal to the algebraic geometry of super vector spaces (that is, a cocommutative Hopf algebra in that symmetric monoidal category, subject to some conditions). An "external group" is a supergroup with a distinguished element (of $G(k)$) of order at most 2. We do not look at all representations of the group, but only ones on which the distinguished element acts as parity (and we only allow distinguished elements such that the adjoint representation is allowed). Deligne's theorem is that every nice category is this subcategory of representations of an external group. We recover the group as automorphisms of the fiber functor to SVect. As you note, the automorphisms of the identity functor of SVect is $Z/2$; that is its external group, with the nontrivial element distinguished. Vect is representations of the trivial group. These are the initial and final external groups, and they are not isomorphic, so external groups are not group objects in some category. In general, internal representations of a supergroup are representations of an external group which is the semidirect product of the two element group acting on the original supergroup. Representations of ordinary algebraic groups reuse the group as the external group, with the trivial element distinguished.

A fairly easy example is that graded vector spaces with the signed symmetry are representations the external group $G_m$, with $-1$ distinguished. Your calculations of the automorphisms of the fiber functor on chain complexes and of the full category of representations of $G=G_m \ltimes G^{0|1}_a$ each imply that the relevant subcategory of its representations are chain complexes of ordinary vector spaces.

This is all based on just the first few pages of Deligne's paper. I don't think he has a name for what I call "external groups"; the Mueger appendix you cite calls them supergroups, but I think that's a bad choice.