In React project: Button to add items to an area in a component is not shown unless the area is defined in the page template.

This is incorrect, the area definition in the component should be used - not the one in the parent page.

For example "nestedAreas" here:

https://git.magnolia-cms.com/users/canh.nguyen/repos/magnolia-react-tutorial/browse/light-modules/sample-light-module/templates/pages/standard.yaml