What will provide this "HUD" exactly? Will it function when running these apps on other platforms, or under other environments?

Menu export over D-Bus, mainly, or anything exported over D-Bus, plus any external scripts or programs relevant to the app.

A robust API (like KDE's DWD) would of course not be supported by most applications, but neither is any particular CSD implementation, and that's OK. But at least applications with different tool-kits could use such an API, unlike Gnome CSD which is just for Gnome apps.

Will it function when running these apps on other platforms, or under other environments?

Yes, HUD implementations are completely independent of the WM under X and relatively independent under Wayland. Gnome menu-less CSDs aren't supported on KDE, not to mention other operating systems.

What problem is it actually solving?

Same problems Gnome's HIG is solving, and many more. The "more" includes consistency, flexibility, discovery etc.

I think I may need to clarify the benefits a little better, but I think it ultimately boils down to some people thinking it would be valuable and other people thinking it's not worth it.