The pullback of the tangent bundle of the sphere by the Gauss map is indeed the tangent bundle of the hypersurface, and the pullback of the Levi-Civita connection is indeed the Levi-Civita connection. Let's make use of the old-fashioned notation of frames: write points of Euclidean space as $x$, and orthonormal frames as $e_1, \dots, e_{n+1}$. Let $e$ be the matrix whose columns are the $e_i$. Take the bundle of adapted frames $FM$ of a hypersurface $M$. The adapted frames are the orthonormal frames with $e_{n+1}$ perpendicular to $M$. Write the soldering 1-forms as $\omega_i=e_i \cdot dx$, and the connection 1-forms as $\omega_{ij}=e_i \cdot de_j$. Similarly write the points of the unit sphere as $X$ and the orthonormal frames as $E_1, E_2, \dots, E_{n+1}$. Let $E$ be the matrix whose columns are the $E_i$. Then the frame bundle of a hypersurface $M$ maps to the frame bundle of the sphere by taking $(e,x)$ to $(E,X)$ in the frame bundle of the sphere, where $E=e$ and $X=e_{n+1}$. We can then see that the connection 1-forms of $M$ are $\omega_{ij}=e_i \cdot de_j=E_i \cdot dE_j=\Omega_{ij}$, matching up the pullback connection 1-forms of the sphere. The tangent bundle is an associated vector bundle, so matching up the connections on the principal bundles ensures that the connections on the associated bundles agree. You have to get used to this old-fashioned notation, but it does unwind to a proof that this particular lift of the Gauss map is connection-preserving. This is surprising, because it says that the curvature 2-form pulls back to the curvature 2-form. But that doesn't force the sectional curvature, or even the scalar curvature, to agree.
