Maybe it would be better if somebody fixed that thing with the menus. But I think IIRC if you wanted transparency in old X11 without a compositor, the toolkit had to copy the region under the popup, and then blit it to the popup... Yes, this meant that content in the transparent region that changed would be stale until the popup went away. So someone would have to decide if that would be a viable solution, or maybe someone could get away with just using the existing XShape stuff in GDK somehow.
As for the solution, why not remove shadows for non-composing wms?
The issue is not the shadows, the popup window itself is shaped. All that black region is supposed to be transparent. To fix with CSS you'd have to get rid of the "arrow" and make the corners square. But that may reduce usability of the popover since it no longer "points" to the parent. Maybe it could be positioned inside the popup somehow.
GtkMenu, GtkMenuBar, and GtkMenuItem are gone so you have to use GtkPopover* instead which has this issue.
I would love to not use adwaita except gtk4 has crippled the ability for users to theme their apps, especially when libadwaita is used. Also gtk3 does not have this issue even with adwaita.