@JasonStarr It's quite a handful of information for me, I'll have a long hard think about it, thank you! Anyway, I think the group of torsors you mentioned above is $H^1(k,\mu_n)$ which has to be $k^\times / (k^\times )^n$, not $k^\times /(k^\times )^2$.

@JasonStarr Also, maybe I should've mentioned this in my post, but the examples cited consider curves of genus at least $1$, do you think it would play a part in making the result hold? Actually we can rephrase the problem as such: does there exist an adelic point $(P_v)$ in the affine curve $X$ that does not survive at least one $X$-torsor under some finite etale $G$, such that its image $i(P_v)$ in $C(\mathbb{A}_k)$ survives all $C$-torsors under ALL finite etale $G$? I'm hoping for a negative answer in the case of genus $\geq 1$ but I'm not sure how to proceed.

@JasonStarr Regarding surviving a torsor, I'm using the notion introduced by Stoll in the paper cited above. Fix a torsor $f: Y \rightarrow X$ under finite etale $G$, then $(P_v)$ survives $f$ if its image in $\prod_v H^1(k_v,G)$ (where the evaluation at each $v$ is given by pull-back of $f$ to a torsor over $\mathrm{Spec}\,k_v$) comes from the localization map $H^1(k,G) \rightarrow \prod _v H^1(k_v,G)$. Is this the same as your definition? I would like to know how is the Brauer set involved in this, otherwise your (counter)example looks completely new to me!

@naf Do you know how the rephrased question would look like? I've look through the literature but there is little to no references that specifically consider $H^1(Y,G)$ for $G$ that is non-abelian. In other words (if I'm not wrong), how do we relate the etale fundamental group to unramified non-abelian coverings of a variety? More specifically, of a smooth projective curve?

@abx I didn't know they would be different, in my examples, one is an open immersion while the other is closed. Could you roughly explain why, or direct me to some references?

@abx so what sufficient condition am I missing for it to be injective? What if $X$ is dense open in $Y$, is it enough? I admit I was hasty in my generalization.

@FelipeVoloch I really appreciate your clarification, for the entire first half of the question I took the notations straight out of your paper. Thank you.

Thanks for the thorough explanation! By Milne's EC, with only the requirement that $C$ is a smooth geometrically integral curve with perfect residue fields $\kappa (v)$ for all closed points $v$, our exact sequence is $$0 \rightarrow \mathrm{Br}(C) \rightarrow \mathrm{Br}(k(C)) \rightarrow \bigoplus_vH^1(G_v,\mathbb{Q}/\mathbb{Z}) \rightarrow H^3(C,\mathbb{G}_m)$$ and so I guess the key lies in the triviality of the ramification map $$\mathrm{Br}(k(C)) \rightarrow H^1(G_p,\mathbb{Q}/\mathbb{Z}),$$ where $\{p\} = \bar{C}\backslash C$. Am I right? Also, how did your $H^3(C,\mathbb{G}_m)$ vanish?

@R.vanDobbendeBruyn Thanks for the link. I tried to follow your construction, but the middle term would be $$H^2(\mathrm{Spec}\,\mathcal{O}_{k,S},j_*\mathbb{G}_m) = H^2(\mathrm{Spec}\,k(C),\mathbb{G}_m) = \mathrm{Br}(k(C))$$ since the residue field of the generic point is the function field of the curve $C$. Here $\Omega_k$ is the set of closed points of $C$, and I set $S$ to be $\Omega_k \backslash\{p\}$. I know I went wrong somewhere, and I don't see how the role of finite fields come into play here.

@Sasha What I meant was, since we needed the requirement of removing closed subschemes of codimension $\geq 2$, the varieties $X$ we are considering cannot include curves, because their only "proper" closed subschemes are finite sets of closed points, which are of codimension $1$. I know of examples where the Brauer group of a smooth projective elliptic curve $C$ is isomorphic the Brauer group of $C$ minus a closed point, so what I'm asking is if there are any examples where the isomorphism of Brauer groups fail to hold after removing a finite set of points from a smooth projective curve.