I've found this article and I am trying to set up Chrome extensions for my organization. https://support.google.com/chrome/a/answer/2663860?hl=en
The guide instructs to navigate to
Device management > Chrome management > User settings > Chrome Web Store Permissions.
However, I cannot find User Settings.
This is what I see under Chrome management:
Has anyone done this before, and could someone please help me to publish a private app?
The screenshot you've linked is of the Apps section for core services, not Device management. That being said, I'm guessing you have not purchased Chrome device licenses so you're probably not seeing the same view as what's being shown in the article you've linked.
In that Apps section, you'll want to go to Additional Google services. There, you'll see Chrome Management at the top of the list and clicking on that will expose the User settings you're looking for.
Please note, this has nothing to do with the actual publication of the app/extension, this is how you'll actually push out the app/extension to users as a policy. Once you publish an app/extension, you'll be able to manage it with these settings.