Create a Product, via the API, which is in POS but not the Website

[CoCreate]

I am creating a special simple product entry screen for a client as a Custom App. 

They create one of a kind products in their workshop (and create the Product in Shopify there), and send them to retail stores, where they use the Shopify POS app to sell the products.

Most of the products are not sold through the website and so the App that I am making must create Products that are not shown on the website.

So - the essence of my enquiry is -

    How can I create a product in Shopify, via the Admin GraphQL API, which is available for POS and Transfers, but will not appear on the website ?

I know this has been asked many times, yet I haven't found a good definitive answer.
For example, I've read, and spoken to the Shopify support about, the published_scope flag, which doesn't allows setting to POS via the API, but in some cases Spotify will allow that for you, if it you do a private app, but private apps are being deprecated, or maybe if the client is on Shopify Plus (which mine is not).
I'm interested to know all of the different ways a product could be flagged to not appear on the website.
Maybe it is possible to create an inventory location that doesn't appear on the website for example??
I'm new to Shopify, so I really do welcome all ways this can be done.

Thank you

[policenauts]

It's either set published to false or published_at to null. I'm not sure which, but I do both and it works.