Nginx location / vs /artifactory

The location directive is /artifactory/ and not / because you are using a public context. That is to say, all access to Artifactory will be in the form of servername/artifactory/ and not servername/. This has the advantage that you can use the same URL for multiple applications, for example, something like this:

Artifactory -> servername/artifactory/Jenkins -> servername/jenkins/My Custom Service -> servername/myapp/

In other words, it allows you to reuse the same servername (and port) with different contexts for different applications. If your reverse proxy was listening at the root level, all the requests would be forwarded to Artifactory.

Now to answer your specific question, why does Artifactory do this? That is likely for clarity/consistency since the default tomcat shipped with Artifactory uses the artifactory keyword for its context. You are of course free to remove the public context from the NGINX configuration and everything will work as expected with the root context servername/, provided you make all the necessary changes (removing it from the rewrites, location, and X-Artifactory-Override-Base-Url).