Another thing worth noting is that the Portal license for anonymous submissions is linked to a specific server so it matters which host you access even if they're all pointed at one database.
If you don't see the option to make it public, then the instance you are accessing does not have the public portal.
However, that doesn't necessarily mean you don't have the license at all, it just means the instance you're accessing does not the license.
As an example, we have two Forms instances; one is internal, and the other is public-facing in the DMZ.
Both of our instances point to the same database, and the routing service is run on the internal instance. However, the Portal license is attached to the DMZ instance for public/anonymous submissions.
As a result, when I access business process settings, I only see the public vs restricted option if I'm logged into the DMZ instance since that is where the license is attached; I believe this is what Angela may have been getting at.
As Abdellah mentioned, you can check LFDS to see if you have the license, and which host has it, so you know where to log in to apply that setting.