So if you want to define $\mathscr{D}(S)$ as a dg-category, you need to first define what it means to take a limit of dg-categories over an index $\infty$-category. It's pretty awkward to mix dg and $\infty$ in this way, and it's easier to just work $\infty$-categorically from the beginning.

@DonuArapura Denis's response above is very good, but let me add one reason why you might choose to work with $\infty$-categories over dg-categories even if you cared only about characteristic zero (derived) objects. Consider the problem of defining the derived category of quasi-coherent sheaves on a derived scheme $S$. One way to do so is to take the limit of $\mathscr{D}(R)$ over all affine (derived) schemes $\operatorname{Spec} R$ mapping to $S$. This index category is an (unstable) $\infty$-category; in particular, it is not a dg category. (cont.)

@EvansGambit To conclude that there is a point in $\mathbb{A}^n$ that does not lie in a given proper subvariety (in this case the union of Yosemite Stan's comment) you need $k$ to be infinite. (Think about the case of the subvariety $x(x-1)\cdots(x-(p-1))=0$ in $\mathbb{A}^1_{\mathbb{F}_p}$.)

Let $p,q$ be points in $U$, and let $r$ be a generic point of $U$. Then the line through $p,r$ and the line through $q,r$ both lie entirely inside $U$ (this is where the codimension $\geq 2$ gets used.)

What is the original problem? Given the information you've provided, my intuition is that this formulation (via systems of equations of matrices) is quite possibly much harder to directly approach than the original problem.

@FedericoCarta Well, that means that finite subgroups of $U(2)$ correspond to $\mathbb{Z}_2$-invariant finite subgroups of $SU(2)\times U(1)$. Every finite subgroup of $SU(2)\times U(1)$ is contained in one of the form $A\times B$, for $A,B$ finite subgroups of $SU(2),U(1)$, respectively. It seems very likely to me that you can use this to give a complete classification.

@Meow This is a very deep question, and a bunch of work (not all published) has been done on it for different values of $X$. I don't know of any individual such $X$ that "best approximates" $\hat{g}\operatorname{-mod}$, but by studying the map for different values of $X$ one can learn a huge amount about $\hat{g}\operatorname{-mod}.$ If you name a specific $X$ you are curious about I can tell you what is known (at least at critical level, where the story is a bit more straightforward).

@Meow Aha, OK, that explains it. When I say Whittaker I mean the operation of taking $(LN,\chi)$-equivariant D-modules, defined e.g. in Section 1.13 of the paper of Raskin I linked earlier.

@Meow Well, my confusion is that you seem to be placing Whittaker on the algebraic side, while I think of it as a purely geometric construction.

@Meow Hmm, what is it about Whittaker itself that you don't like? In any case, here is one possible answer: For $X$ the affine flag variety (or the affine grassmannian,) Whittaker is the same as what is called "baby Whittaker," which is what is studied in e.g. Arkhipov-Bezrukavnikov (arxiv.org/abs/math/0201073).

@Meow Oh, if that's what you want then you're in luck: Whittaker is actually easier to describe on $D(X)$ than on $\hat{g}\operatorname{-mod}$. I more meant that there's no reason to take $X$ specifically the semi-infinite flag variety.

P.S. The semi-infinite flag variety is an important part of this story, but I think for your question specifically it is a red herring. It appears when you think about $LN$ invariants without a twist. With a twist, the relevant geometry is instead that of Whittaker. This might seem like a small difference but it's really not.

A more algebraic answer is given by A.2 of the cited paper. Raskin's heuristic explanation is that it is "cohomology along $\mathfrak{n}[[t]]$ and homology along $\mathfrak{n}((t))/\mathfrak{n}[[t]],$ and his definition makes this idea precise. My personal way of thinking about it is that it is "ordinary Lie algebra cohomology for $\mathfrak{n}((t))$, but shifted by $\operatorname{dim}\mathfrak{n}[[t]]$ degrees," which is complete nonsense (because $\mathfrak{n}[[t]]$ is infinite dimensional) but captures its behavior.

A very satisfying answer is provided by arxiv.org/abs/1611.04937. In short, the Drinfeld-Sokolov functor comes from the 2-categorical functor of Whittaker coinvariants, applied to the category of Kac-Moody modules. However, this answer may be too categorical for your taste.