Your final request, even if augmented to require the locally closed subvariety $V$ to be smooth, is false over every field $k$ of characteristic 2 and 3, with counterexamples of any rank $n \ge 1$ over any $k$ of characteristic 2, with $V \subset G$ of very large dimension (unbounded as $n$ grows). So avoiding finite fields of bounded size is insufficient in characteristics 2 and 3. A source of counterexamples is certain "well-known" exceptional (non-central) isogenies through which Frobenius non-trivially factors, and these isogenies only exist in such characteristics.

I will describe counterexamples with $G = {\rm{Spin}}_{2n+1}$ for any $n \ge 1$, and a variant of the same ideas (but carried out by more conceptual methods based on the structure theory of semisimple groups rather than based on manipulations in linear algebra) gives counterexamples for ${\rm{Sp}}_{2n}$ ($n \ge 1$) and ${\rm{F}}_4$ over any field of characteristic 2 and for G$_2$ over any field of characteristic 3; see section 7.1 of the book "Pseudo-reductive groups" for a unified development of the relevant general principles (but again, the phenomenon at hand has been widely known for many decades).

Let $V = k^{2n+1}$ and $$q = x_0^2 + x_1 x_2 + \dots + x_{2n-1} x_{2n}.$$ The associated symmetric bilinear form $B_q(v,v') = q(v+v') - q(v)-q(v')$ has defect space $V^{\perp} = \{v \in V\,|\,B_q(v,\cdot) = 0\}$ that is a line since ${\rm{char}}(k)=2$ (this line is $k e_0$). On $V/V^{\perp}$, the induced bilinear form $\overline{B}_q$ is symplectic since ${\rm{char}}(k)=2$. Thus, we get a natural composite homomorphism $$f: G = {\rm{Spin}}_{2n+1} \rightarrow {\rm{SO}}_{2n+1} \rightarrow {\rm{Sp}}(\overline{B}_q) = {\rm{Sp}}_{2n}.$$

Let $T$ be a split maximal $k$-torus in $G$, and let $\Delta$ be the base of $\Phi(G, T)$ associated to a choice of positive system of roots $\Phi^+$. The multiplication map $\prod_{a \in \Delta} \mathbf{G}_m \rightarrow T$ defined by $(\lambda_a) \mapsto \prod a^{\vee}(\lambda_a)$ is an isomorphism since $G$ is simply connected. In particular, ${\rm{Lie}}(T)$ is the direct sum of the "coroot lines" ${\rm{Lie}}(a^{\vee}(\mathbf{G}_m)) = {\rm{Lie}}(a^{\vee})(\partial_t|_{t=1})$.

The kernel $\mathfrak{n} := \ker {\rm{Lie}}(f) = {\rm{Lie}}(\ker f)$ is an ${\rm{Ad}}_G$-stable subspace of $\mathfrak{g}$, and it is nonzero and proper. Explicitly, it is the direct sum of the root lines for the short roots and the coroot lines ${\rm{Lie}}(a^{\vee}(\mathbf{G}_m))$ for the coroots associated to the short roots in $\Delta$. (In fact, $\mathfrak{n}$ is the unique minimal non-central ${\rm{Ad}}_G$-stable subspace of $\mathfrak{g}$.) Hence, $\mathfrak{n} = {\rm{T}}_e(V)$ for the smooth locally closed subvariety $V \subset G$ through the identity $e$ given as the closed subvariety of the open cell associated to $\Phi^+$ by forming the direct product (embedding via multiplication into $G$) of the $T$-root groups for the short roots and the direct factors $a^{\vee}(\mathbf{G}_m)$ for short $a \in \Delta$.