This is basically an instance of "spreading out" principles in algebraic geometry, combined with arguments specific to simply connected semisimple groups over general fields away from a few low-rank examples over fields of size 2 and 3. The formulation of the question is a bit imprecise, since the notation $G_v$ isn't really defined for all $v$, and making precise what it means for all but finitely many $v$ implicitly entails many of the steps below. We'll work directly with group schemes over rings since that is the natural framework to systematically discuss and analyze all questions of this type.

First, the spreading out. Let $R$ be a domain with fraction field $K$ (for example, $K$ a global field and $R$ a ring of $S$-integers for some non-empty finite set $S$ of places of $K$ containing the archimedean places) and $G$ a connected reductive $K$-group. By general spreading-out results in EGA IV, there exists a nonzero $r \in R$ such that $G$ is the generic fiber of a smooth affine $R[1/r]$-group scheme $\mathscr{G}$ whose geometric fibers are connected; we may and do rename $R[1/r]$ as $R$.

By Corollary 3.1.11 of the article "Reductive group schemes" in the Proceedings of the 2011 Luminy summer school on SGA3 (applied to $K$ viewed as the direct limit of $R[1/r]$'s) we can replace $R$ with $R[1/r]$ for some nonzero $r \in R$ to arrange that all geometric fibers of $\mathscr{G}$ over ${\rm{Spec}}(R)$ are reductive, and even semisimple when $G_{\overline{K}}$ is semisimple.

There is a finite separable extension $K'/K$ such that $G_{K'}$ is split (as $G_{K'_s}$ is split). Fix an $R$-finite domain $R' \subset K'$ with fraction field $K'$, so $K'$ is the direct limit of subrings $R'[1/r]$ for nonzero $r \in R$ (as $R' \otimes_R K = K'$). Hence, by inverting some such $r$ (again using the EGA IV spreading-out principles, for properties such as being an open immersion) we can arrange that $\mathscr{G}_{R'}$ admits an "open cell" with fiberwise maximal split $R'$-torus and root groups over $R'$ recovering the usual ones on the generic fiber. That already tells us the root datum on every fiber, and the root datum detects being simply connected.

Thus, now assuming $G_{\overline{K}}$ is semisimple, absolutely simple, and simply connected (all detected by the root datum: irreducible root system and coroots spanning the character lattice of a maximal torus), it follows that the same holds for all fibers of $\mathscr{G}$ over ${\rm{Spec}}(R)$.

In the special case that $K$ is a global field and we did spreading-out relative to some ring of $S$-integers, by enlarging $S$ we see that for each $v \not\in S$ the pullback $\mathscr{G}_{O_v}$ over the $v$-adic completion of $O_{K,S}$ is a smooth affine $O_v$-group. Hence,
$\mathscr{G}(O_v) \rightarrow \mathscr{G}(\mathbf{F}_v)$ is surjective. The role of GL$_n$ in the setup is subsumed in the theory of group schemes over rings (namely, again by enlarging $S$ depending on some initial inclusion of $G$ into some ${\rm{GL}}_n$ we ensure that the schematic closure of $G$ inside ${\rm{GL}}_n$ over the $S$-integers coincides with $\mathscr{G}$ as an $O_{K,S}$-group).

In other words, the crutch of GL$_n$-embeddings is just a device to discuss flat affine integral model group schemes (of finite type) without speaking in terms of schemes. But it is precisely the **relative smoothness** of such $\mathscr{G}$ (or more concretely of its Zariski closure in such a GL$_n$ over $R$ at the cost of passing to some $R[1/r]$) which guarantees that the "ad hoc" definition of $G(\mathbf{F}_v)$ in the question actually agrees with the more conceptual notion $\mathscr{G}(\mathbf{F}_v)$ (of $\mathbf{F}_v$-points of the $O_{K,S}$-group scheme $\mathscr{G}$).

Consequently, we can now **forget about** the entire setup with group schemes over rings of integers (that has already served its purpose, bringing us to a suitable $R = O_{K,S}$ over which we've performed enough nice spreading-out)
and are reduced to problem entirely about connected semisimple groups over fields: if $H$ is a connected semisimple group over a field $F$ such that $H$ is absolutely simple and simply connected, then away from a few low-rank cases over $F$ of size 2 or 3, prove $H(F)$ is perfect and that $H(F)/Z_H(F)$ is a simple finite group (with $Z_H$ denoting the center of the $F$-group $H$). This is an application of the role of Tits systems in the Borel-Tits structure theory for connected reductive groups over fields, and is discussed by a variety of methods in a variety of references (depending on your taste).