Hey, I just went through the same issue. I think it is because I manually added a Variant to the “Gift Card” product via “Products”, rather than going through the “Gift Card> Add a Gift Card” method. After I did this, my gift card product started showing the out of stock.
I looked through shopify docs but was unable to find a way to actually update an existing product to add more variants:
https://help.shopify.com/en/manual/products/gift-card-products/add-update-gift-card-products
While it says “update” is an option, I was unable to do so.
To resolve the issue on my end, I recreated the gift card product with all the “Denominations” I previously had, plus the additions I was looking to make in the first place. I tested this out and it no longer has the Out Of Stock issue at checkout.
I then moved the old one to “Draft” status and renamed it with an “- OLD” at the end of the name. Not sure if this is necessary but instead of deleting the product entirely, I wanted to ensure that the customers who bought these would still be able to use it.