On a product in Store in the Magpie app, the Status choice is replaced by what the website offers. A product that is not live has Make live, which applies the same publishing checks. Remove from store and Archive product are in the product's menu and ask you to confirm: a product removed from the store becomes a draft and keeps its orders, and an archived product's existing orders keep their records.
Make live, Remove from store and Archive product in the Magpie app
A product in the Magpie app has Make live while it is not live, and Remove from store and Archive product in its menu, which ask you to confirm.