It is a thing, it’ll adjust the background color if you use it in your theme. It inherits from paper-list-background-color.
It’s a color CSS style. Not a background css style.
There is not, it’s fragmented across the lovelace codebase.
To be honest, I can’t believe your original code ever worked. The variable name insinuates that it’s a color only and not a full css background object. You can attempt to use sidebar-background or ha-sidebar-background but I doubt it will work.
When inspecting the element, it also looks like card-mod can adjust it. So that might be something that could work.