Since begining 2026 Shopify updates product status with a new status named Unlisted.
Here are some details copied from what Unlisted status differs from other two status:
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We’ve introduced a new product status called “Unlisted”. It hides products from store search and collections, all sales channels, internet searches and Shopify Catalog—while still allowing access via a direct URL. It’s ideal for private sales, warranties, add-ons, or “free gift with purchase” items you don’t want discovered or used outside of their intended purpose.
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This should work as Active and it should update stock so is not exactly an idea but something that Developers should take care and update their integrations accordingly with Shopify updates.
Hope it helps and raises conscious of the potential of the new Shopify status.
It was added a bit ago - you need to make sure that on the Shopify connector 'Only Sync Active Products (Exclude Draft)' is set to OFF
Also while I'm at it, the name of the above needs a bit of tweaking as to my mind, it should reference the inventory sync aspect as to me it implies it's only a product download sync. Maybe something like 'Sync Active Products & Inventory Levels (Exclude Draft)'