I meant something more like "up to isotopy" than "up to diffeomorphism"; e.g. in the case $n = 2$, $L$ is an actual link in $S^3$, and I certainly want to remember more than the number of connected components. (I have a vague recollection of seeing a more canonical definition of the link, avoiding the dependence on $\epsilon$). Your counterexample then shows that the most we could hope for is that the isotopy class of the link determines the deformation-equivalence class of the singularity.

Oh sorry, I was being careless! The definition you give is naturally the right adjoint to restriction (in the category of continuous representations). However, since the category of continuous representations of a compact Lie group is semisimple (i.e. there’s always a unitary structure), you can apply Frobenius reciprocity to the duals and then dualize back to get the other universal property. This isn’t as natural as the other statement, because it’s relying crucially on semisimplicity - for more general groups, the right and left adjoints are indeed sometimes different.

By unwinding the Frobenius reciprocity theorem, which says the functors of restriction and induction are adjoint, you’ll get exactly this sort of universal property. However, some care is needed - since this definition of induced representation imposes a continuity condition, it will only have the universal property among continuous representations.

I can't find a reference right now but the argument is: The category of $G$-torsors is the same thing as the category of faithful exact tensor functors from the category of representations of $G$ into vector bundles. So a $G$-shtuka is the same thing as a collection of $GL(V)$-shtukas for each representation of $V$ which are compatible with tensor products etc. Thus if I had a $G$-shtuka with one leg, all of the corresponding $\GL(V)$ shtukas would be trivial, and thus the original $G$-shtuka must be as well.

For $\mathrm{GL}_n$, this should come from the fact that pulling back a vector bundle $\mathscr{E}$ on $X_S$ by $F = 1 \times \mathrm{Frob}_S$ doesn't change the degree, so an everywhere-defined morphism $F^*\mathscr{E} \rightarrow \mathscr{E}$ that only vanishes at one point must be an isomorphism. For general groups, I think you can just use the above argument via the Tannakian description of torsors.

I think the problem of constructing perverse sheaves on the stack of G-bundles or shtukas which recover a given eigenform is a hard problem, core to the difficulties in the geometric Langlands program. In V. Lafforgue's proof of the Langlands conjectures for function fields, he sidesteps this issue by noting that $H^0_c(\mathrm{Sht}_0, \overline{\mathbf{F_q}}, \mathbf{Q}_\ell)$ may be identified with the set of all cuspidal automorphic forms, where $\mathrm{Sht}_0$ is the stack of shtukas with no legs. Then you can build the Galois reps by looking at cohomology of IC ("constant") sheaves.

For what it's worth, the hypothesis that $P$ and $\mathscr{P}$ are parabolic is only used in the proof of point 1. above. In general, this conversation should carry over to the case that $H \subseteq G$ is a (connected, smooth) closed subgroup to show an equivalence between the category of $N_G(H)$-torsors and the category of pairs $(\mathscr{E}, \mathscr{H} \subseteq \mathrm{Aut}(\mathscr{E})$ where $\mathscr{H}$ is an "inner form of $H$" in the above sense.