5. Check the published site's sharing settings are set to public
If your site redirects to the sites.google.com URL or asks visitors to sign in then your published site's sharing settings are not set to public - here is how to make your new Google Site public to anyone on the internet:
Open your new Google Site for editing
Use the Share with others button (looks like a person with a + )
In the Links section use the Change... link
For the Published site link choose the Public option and use the Done button.
If you can't see the Public option for the Published site then the most likely scenario is you use Google Workspace and your Google Workspace admin has restricted public sharing - you need to talk to your Google Workspace admin on how to change this - see Set Drive users' sharing permissions.
If you do want your site to be private then the custom URL will always redirect to the sites.google.com URL - this is by design.