Let $G$ be a split semisimple algebraic group and let $C$ be a curve of genus $g$ over $\mathbb F_q$. Assume $g \geq 2$.

The number of $\mathbb F_q$-points of $\# \operatorname{Bun}_G(C)$, where each point is weighted by the inverse of the order of its automorphism group, is $(1+o(1))q^{ (g-1) \dim G}$. This follows from the Tamagawa number 1 theorem, though I think this basic estimate is easier to prove.

Is this still true if we do not divide out by automorphisms? Obviously not. If $G = SL_2$ we can take $L + L^{-1}$ for infinitely many line bundles $L$ to get infinitely many different isomorphism classes of vector bundles with determinant $1$. But it is true if we restrict to stable vector bundles, because the number of automorphisms is bounded.

Let $U$ be an open subset of $\operatorname{Bun}_G(C)$ that is of finite type. For instance, if $G=SL_n$ we can take the set of determinant $1$ vector bundles $V$ such that $V \otimes L^{-1}$ has no sections for a line bundle $L$ of large degree.

Is the number of isomorphism classes of $G$-bundles on $C$ contained in $U(\mathbb F_q)$, not weighted by automorphisms, still $(1+o(1)) q^{(g-1)\dim G}$? Or does the number of points on $\operatorname{Bun}_G$ jump when you add even midly unstable bundles?

I promise I have a really good reason to care about this "evil" count.

I think I can prove this estimate in the case $G=SL_2$. As I already mentioned stable vector bundles are fine. The remainder are split sums $L + L^{-1}$, the number of which with $L$ of degree $n$ is $(1+o(1)) q^g$, and nontrivial extensions of $L^{-1}$ by $L$, which are classified by $H^1(L^2)$, so they are cardinality of them is at most the cardinality of $H^0 ( K L^{-2})$. Summed over all line bundles $L$ of degree $n$, this is at most a constant times the total number of nonzero sections of all line bundles of degree $2g-2-2n$ up to scaling, which is equal to the number of effective divisors of that degree, or $q^{2g-2-2n}$. As $n \geq 0$, either one is less than $q^{3g-3}$.