want changes to submenu applied to all submenus

Also, it is not clear from exactly where I should do that. From my mouse on the offending menu? Mouse on one of the non-submenu items in the offending menu? Mouse on one of the submenus on the offending menu?

No matter what item you click. Customize menu is applied to the currently opened menu. You can check the "top" menu in the "Menu level" field. The "level" for top menu contains the name.

If you right click the toolbar and select "Customize menu" you'll see the levels only and no "This menu:..." in the menu level list.

But, no matter where my mouse is when I right click to get Customize Menu, checking the box to overwrite the submenus does not reset their color schemes.

Maybe you need to tun on the option "Remove the menu levels in sub-menus". This option removes the levels definitions from the submenus. This can be the cause why some submenus still have own colors.