Use fenix toolchain in nix shell (#18227)

In #17974 we explicitly depend on rustc/cargo for the nix devShell,
however the fenix overlay that contains the latest stable versions was
not being applied to that shell. This led to the shell inheriting
whatever rustc/cargo was on nixos-unstable from nixpkgs, which sometimes
lags behind. This change fixes that, and also restructures the flake to
ensure that all outputs rely on the overlaid `pkgs`.

Release Notes:

- N/A