Suppose a scheme $X$ has tame quotient singularities. Does there exist a smooth DM stack $\mathcal X$ with coarse space $X$ so that the coarse space morphism $\mathcal X\to X$ is an isomorphism away from codimension 2?

If $X$ is finite type over a field, the answer is yes, by the following argument. Theorem 4.6 of Fantechi-Mann-Nironi's Smooth toric DM stacks shows that such stacks have a universal property, so it suffices to show that they exist étale-locally (the universal property ensures the locally constructed canonical stacks will glue). Étale locally, we may assume $X=U/G$, where $U$ is smooth and $G$ is a finite group acting tamely on $U$—that's the definition of "tame quotient singularities". Let $H\subseteq G$ be the (normal) subgroup acting by pseudoreflections (through a given closed point $x\in U$; shrinking $U$ if necessary). Then by the Chevalley-Shephard-Todd theorem, $T_x/H$ is smooth, where $T_x$ is the tangent space at $x\in U$. If $X$ is defined over the residue field $k(x)$, then we can construct an $H$-equivariant morphism $U\to T_x$ sending $x$ to the origin, which is étale at $x$. Since $T_x/H$ is smooth, so is $U/H$. (If $X$ isn't defined over $k(x)$, base change to $k(x)$ and get smoothness of $U/H$ by descent.) The action of $G/H$ on $U/H$ is free away from codimension 2, so $\mathcal X = [(U/H)/(G/H)]$ does the trick.

As far as I can tell, the only place we had to use that $X$ is defined over a field was in showing smoothness of $U/H$. Without it, we can't construct the étale morphism $U/H\to T_x/H$ we need to deduce smoothness of $U/H$ from smoothness of $T_x/H$.

Can this "over a field" condition be relaxed to get the existence of canonical stacks in an absolute setting (i.e. over $\mathrm{Spec}(\mathbb Z)$)?

Context: my more general goal is to understand if canonical stacks exist over an arbitrary base $S$. That is, suppose $X$ is a scheme (probably locally of finite type) over a base $S$ which has an étale cover by a disjoint union of schemes of the form $U/G$, where $U$ is smooth over $S$, and $G$ is an abstract group acting on $U$ (over $S$), with order relatively prime to all residue fields of $U$. Then does there exist a stack $\mathcal X$ which is smooth over $S$ and has coarse space $X$, such that the coarse space morphism is an isomorphism away from codimension 2 (on both $X$ and $\mathcal X$)?