Decided to try this feature out.
Didn’t realise you had to specifically state who can access.
However the email input is disabled and I’m getting an error on the check below.
(Android 17 on Pixel 10 Pro)
In addition on web (e.g. desktop), sharing just shares the link, it does not pass you to a form where you can select privacy options!
As a result I’m unable to publish my site for public consumption 
Here is the guidance:
Source: https://help.openai.com/en/articles/20001339-creating-and-managing-chatgpt-sites
Note it claims this is available on ChatGPT web, but I do not see this option - Share does not lead to “Who has access”
This sharing button does not lead you to a form, it only shares the URL which remains inaccessible because I’ve not been able to set up the sharing options.
I tested this using my personal account in the iOS app, and the option is available:
Interestingly, the site initially required ChatGPT authentication even though I had selected “Anyone with the link.” I then asked ChatGPT to remove the authentication requirement and make the site publicly accessible, which resolved the issue.
Please give this a try when you get the option to manage access, and let me know whether it works for you as well.
Also flagging this with the team.
Still fails - I get the same screenshot as in the OP, but nice to see it works on iOS & your project
I wonder if it is a project specific glitch - I might try another @sites effort to see if I can reproduce.