[EDIT2 (oops made mistake)][EDIT2]: I just computed the $G_2<\mathfrak{gl}_{14}$, $p=2$ example in GAP and it's telling me that its normaliser is dimension $22$ and its centraliser is dimension $1$ ??? Leave this with me]
AndSo your proposed surjection breaks for $G_2$ in characteristic $2$.
But verily I get your proposed surjection for $p=3$ and(and $p=5$ which is a good prime but just to make sure).
I even checked to make sure that $\mathfrak{pgl}_4$ has no outer derivations when $p=2$ and this does seem to be correct. I would have expected there to be just one extra outer derivation for $\mathfrak{psl}_4=Lie(G_2)$, but clearly these are not linearly equivalent in whatever appropriate sense. BTW The computerThis must have a cohomological interpretation---probably some version of the $7$-dimensional module for $G_2$ is not having anyturning up.]
[EDIT4 According to GAP, the simply connected version of it when calculating $F_4$ has no outer derivations in char 2 (where it is not simple) or 3 (where it is simple so adjoint and s.c. are isomorphic) and $E_6$ has no outer derivations in char 2 or 3 (again, it's simple so adj=sc), but I would guess it might take a day or two to complete $E_7$ and $E_8$. If you want, you can email me and I can do these for you.]