Published and shown publicly are related, but they are not the same setting.
Publishing makes the artwork eligible to appear publicly
A Draft stays private. When you publish an artwork, Studio Operations normally turns public visibility on at the same time.
If Publish is blocked, use Why can I not publish this artwork?.
A Published artwork can still be hidden
Your works → open the artwork → Price → Where and how it sells
Untick Show this work on my website and save. The artwork remains Published in Studio Operations but disappears from the public Work catalogue.
Showing it again does not require republishing
Tick Show this work on my website and save again. Use Show a hidden published artwork on your website for the exact steps.
Archived work is also excluded
A work must be Published + publicly visible + Active to appear in the public catalogue on your Studio Operations Microsite or standalone outside-site gallery.
If all three are correct and the work is still missing, use Why is my artwork not showing in the public gallery?.