Hello,
I don't use that form to feed GMC, but I know they have a subscription plan so support is included.
Check in Merchant Center that the import is periodic, even every night for example. Maybe the incorrect availability depends on that.
If the product has gtin code, yes you need to add it. if you don't have it use the existing identifier attribute [identifier_exists] to indicate that there are no identifier product codes (UPI) available for the product. Identifier product codes are submitted using the GTIN [gtin], MPN [mpn], and brand [brand] attributes.
You should add GTIN in Prestashop product page, you can do it by creating a csv file and then importing it into Prestashop.