Since it is not possible to assign ‘New’ in bulk from ‘quick edit’ on ‘all products’ page creating the ‘New’ label, would it be possible to create a label for a certain tag in order to be displayed on products? For example a tag named ‘New Product’ would display a label ‘New Product’.
You can create an attribute and assign to each product you need or alternatively, you can make products “Featured” and rewrite this label as per your needs if you do not use “Featured” as per its original application.