Hello,
I would like to show a custom label for products with 0 quantity, stock management at product level enabled and backorders are allowed with notifying the user, these are my products settings, and I would the user to notice that the product is not available at the moment in stock and can order only via backorder from product grid/list view.
You can enable the Manage stock and when the product is of stock, the appropriate label would appear.
You can create an attribute with your notifications and activate the label of the attribute, navigate to Products > Attributes > enter the attribute and disable “Show attribute label on products”
You will have to configure all manually. If it does not work for you, try to find a plugin as the default Woocommerce does not provide the option.