Take a look at https://support.google.com/a/answer/6309862 - it may be that you've purchased workspace licenses that support Gmail, but have not assigned one of those licenses to your account.
Thankfully this has been fixed. You can now choose "Transfer accounts" (from the menu in the top right corner) to copy everything to Google Authenticator on a new (or backup) phone.
As a Workspace customer you should also be able to contact Google Workspace support by following: https://support.google.com/a/answer/1047213