I am just posting my final comment as one answer. As the OP notes, there are answers that go through the classification, and there may be a best answer that uses very little. The answer here relates the post to the theory of the automorphism group schemes of semisimple algebraic groups.

For a pinned semisimple group scheme $G$ over a base scheme $S$, there is an associated $S$-scheme $A$ over $S$ representing the function of automorphisms of $G$ as an $S$-group scheme, and $A$ is itself a linear algebraic group that is an extension of a quasi-finite group scheme by a semisimple group scheme. All of this is part of Theorem 7.1.9 of Brian Conrad's excellent notes.

Brian Conrad
Reductive Group Schemes http://math.stanford.edu/~conrad/papers/luminysga3.pdf
(I will add MathSciNet information when I can access that next.)

In particular, the normal factor $G/Z_G$ of inner automorphisms is preserved by isogenies of semisimple group schemes. So an isogeny of group schemes preserves the automorphism group scheme if and only if it preserves the outer automorphism group scheme. For split, pinned semisimple group schemes, this is a group scheme of automorphisms of the associated root datum. This typically is not preserved by all isogenies, e.g., the outer automorphism group of $\textbf{SL}_n\times \textbf{PGL}_n$ is strictly smaller than the automorphism group schemes of $\textbf{SL}_n\times \textbf{SL}_n$ or $\textbf{PGL}_n\times \textbf{PGL}_n$.

As this example illustrates, for the quotient isogeny from the simply connected form to the adjoint form, the group of automorphisms of the associated root datum is preserved, as can be checked by working on field-valued points.

So now the strategy is as follows. By hypothesis, the group scheme under consideration is a form of a split, pinned, semisimple group scheme over $S$. Thus it gives a torsor for the automorphism group scheme of the split form. By forming the quotient by the center, this gives a torsor for the automorphism group scheme of the associated adjoint form. Since the map of automorphisms group schemes from the simply connected form to the associated adjoint form is an isomorphism, this torsor is equivalent to a torsor for the automorphism group scheme of the associated simply connected form of the split, semisimple group scheme. This torsor is equivalent to a form of that simply connected, semisimple group scheme. That form is the "universal cover" of the original (not necessarily split) semisimple group scheme over $S$.

I am just posting my final comment as one answer. As the OP notes, there are answers that go through the classification, and there may be a best answer that uses very little. The answer here relates the post to the theory of the automorphism group schemes of semisimple algebraic groups.

For a pinned semisimple group scheme $G$ over a base scheme $S$, there is an associated $S$-scheme $A$ over $S$ representing the function of automorphisms of $G$ as an $S$-group scheme, and $A$ is itself a linear algebraic group that is an extension of a quasi-finite group scheme by a semisimple group scheme. All of this is part of Theorem 7.1.9 of Brian Conrad's excellent notes.

Brian Conrad
Reductive group schemes.
Autour des schémas en groupes. Vol. I, 93–444, Panor. Synthèses, 42/43, Soc. Math. France, Paris, 2014.

In particular, the normal factor $G/Z_G$ of inner automorphisms is preserved by isogenies of semisimple group schemes. So an isogeny of group schemes preserves the automorphism group scheme if and only if it preserves the outer automorphism group scheme. For split, pinned semisimple group schemes, this is a group scheme of automorphisms of the associated root datum. This typically is not preserved by all isogenies, e.g., the outer automorphism group of $\textbf{SL}_n\times \textbf{PGL}_n$ is strictly smaller than the automorphism group schemes of $\textbf{SL}_n\times \textbf{SL}_n$ or $\textbf{PGL}_n\times \textbf{PGL}_n$.

As this example illustrates, for the quotient isogeny from the simply connected form to the adjoint form, the group of automorphisms of the associated root datum is preserved, as can be checked by working on field-valued points.

So now the strategy is as follows. By hypothesis, the group scheme under consideration is a form of a split, pinned, semisimple group scheme over $S$. Thus it gives a torsor for the automorphism group scheme of the split form. By forming the quotient by the center, this gives a torsor for the automorphism group scheme of the associated adjoint form. Since the map of automorphisms group schemes from the simply connected form to the associated adjoint form is an isomorphism, this torsor is equivalent to a torsor for the automorphism group scheme of the associated simply connected form of the split, semisimple group scheme. This torsor is equivalent to a form of that simply connected, semisimple group scheme. That form is the "universal cover" of the original (not necessarily split) semisimple group scheme over $S$.