Oh okay I see what happened. When I was looking at the "preview mode" text on the side with the link above it I jumped to conclusions and thought that was a preview link. Without really reading the link I just thought it made sense due to their proximity and that I hadn't published yet.
As for the visibility link, I see it now. The issue there is my intuition wouldn't have thought that'd be in the middle of a form. Was honestly expecting it to be something "broken out" and a colored but since it's actionable (with I suppose the dropdown arrow to the side with the other options). My personal opinion for the best place would be in the preview area. Up above the preview like the singlelink is now I'd have a dropdown with the publish options, a separate "Publish" button under that (colored to draw attention), and then a hyperlink of your singlelink with a clipboard symbol next to it to copy the link.
Also under the settings it'd be nice to have a checkbox for a round photo or a square photo.
Edit: After playing with it more I see that it auto-updates/publishes the changes you're making. Due to that I agree that it should publish by default. A staging feature might be a good option for your paid plans.