I don't think you've made an error.  There's a version of Tannakian reconstruction for merely monoidal categories with fiber functors.  Namely, a finite dimensional Hopf algebra is the same thing as a finite tensor category together with a monoidal functor to vector spaces.  See [Theorem 5.3.12 of the Tensor Categories](http://www-math.mit.edu/~etingof/egnobookfinal.pdf).  This reconstruction is compatible with the one for groups, in the sense that the Hopf algebra you recover is $k[G]$.  However, for the Hopf algebra one you look at tensor endomorphisms instead of tensor isomorphisms.  If you start with H-mod for a Hopf algebra H and look at $\mathrm{Aut}^\otimes(F)$ you'll end up with the group G of grouplike elements of H, and Rep(G) will not be the same as H-mod because H is not spanned by its grouplike elements.