When running the mutation publishablePublish, you will get back an error:
PublishablePublish access denied.
This is due to the missing App scope ‘write_publications’. In the Shopify App Permissions console, write_publications is not a permission that can be added.
due to this limitation, when adding/modifying a product by the GraphQL API, you cannot add the product to channels - all products will need to be manually applied to channels via the shopify admin UI
I’ve seen workarounds on the forums that state to use the deprecated methods - but the deprecated methods do not allow the modification of channels besides “online store”