With Airtable it's trivial to hide products when you run out of stock. First, ensure that your Shopify to Airtable sync is set up to sync inventory information from Shopify. When you have that set up, create an Airtable formula column with the below formula:
IF(Inventory>0, NOW(), "")
Now set up this new, formula column to be synced to the Shopify "published_at" Product field. And voila. Now when you run out of inventory for a product, Shopify will automatically unpublish the product from your store. If you add more inventory, then the product will once again show up in your store.